Understanding Broad Mg ii Variability in Quasars with Photoionization: Implications for Reverberation Mapping and Changing-look Quasars

Following previous works in a spherically symmetric geometry (Korista & Goad 2000, 2004), we assume that the density of the clouds covers a broad range of 7 ≤ $\mathrm{log}\,n({\rm{H}})$ [${\mathrm{cm}}^{-3}$ ] ≤ 14 with 0.125 dex spacing. The upper limit is subject to model uncertainties (Ferland et al. 2013), and the lower limit is determined by the absence of forbidden lines in the BLR. An ionizing flux range of 17 ≤ log Φ(H) [${\mathrm{cm}}^{-2}$ ${{\rm{s}}}^{-1}]$ ≤ 24 with 0.125 dex spacing is chosen. The lower limit is related to the sublimation temperature (∼1500 K) of dust grains, which corresponds to a hydrogen-ionizing flux of Φ(H) ∼ 10^17-18 ${\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ (Nenkova et al. 2008). The upper limit is unimportant since the emissivity for most lines is low at large Φ(H). The ionization parameter is defined as the ratio of hydrogen-ionizing photon Q(H) to total hydrogen density n(H):

$$\begin{eqnarray}&&{U}_{{\rm{H}}}\equiv \displaystyle \frac{Q({\rm{H}})}{4\pi {R}^{2}n({\rm{H}})c}\equiv \displaystyle \frac{{\rm{\Phi }}({\rm{H}})}{n({\rm{H}})c},\end{eqnarray} \tag{ 4 }$$

where R is the distance between the central ionizing source and the illuminated surface of the cloud, c is the speed of light, and Φ(H) is the flux of ionizing photons. Constant U_H values are thus diagonal lines in the density–flux plane of Figure 2 (in log-log scale) ranging from 6 (upper left) to −6 (lower right). For each cloud (represented by a point in the density–flux plane), we assume a hydrogen column density N_H  =  10²³ ${\mathrm{cm}}^{-2}$ , abundance Z = Z_⊙, and the same radio-quiet AGN SED (Mathews & Ferland 1987) to perform the photoionization calculations using CLOUDY (version 17.01, Ferland et al. 2017). In addition, we assume an overall covering factor of CF = 50%, as adopted in Korista et al. (2004). Note that the incident continuum does not include the reprocessed emission from other clouds, nor do we consider the effects of cloud–cloud shadowing or continuum/line beaming.

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We obtain the photoionization results on a grid of the density–flux plane, which include 3249 model calculations in Figure 2, where the contour represents the line equivalent width (EW, with respect to the incident continuum at 1215 Å, for direct comparisons with earlier photoionization work) starting from EW = 1 Å in the upper left to ∼1000 Å in the lower right. The EW is directly proportional to the continuum reprocessing efficiency for each line (see Figure 4 in Korista et al. 2004). For constant flux, the vertical y-axis in Figure 2 is equivalent to the radius axis. For all three lines, the most efficiently emitting regions are located in the lower right half, whereas the upper left half are regions of Comptonization (the clouds are transparent to the incident continuum). The EW peaks are marked by black triangles, while the stars correspond to the old standard BLR parameters (Davidson & Netzer 1979). The peak locations are determined by atomic physics and radiative transfer within the large range of line-emitting clouds. We can see that qualitatively the most efficient Mg ii-emitting clouds are slightly more distant than those for the Balmer lines given Φ(H) ∝ L_ion/r² (hereafter L/r²).

Next, we compute the overall line luminosity by summing over all grid points with proper weights determined from assumed distribution functions. Following Baldwin et al. (1995), the observed emission-line luminosity is the integration given by

$$\begin{eqnarray}&&{L}_{\mathrm{line}}\propto \int {\int }_{{R}_{\mathrm{in}}}^{{R}_{\mathrm{out}}}{r}^{2}F(r)f(r)g(n){dndr},\end{eqnarray} \tag{ 5 }$$

where F(r) is the emission-line flux of a single cloud at radius r, and f(r) and g(n) are the assumed cloud covering fractions as functions of distance from the center and gas density, respectively.

We sum the grid emissivity in Figure 2 along the density axis at each radius to obtain the radial distribution of surface emissivity for different lines, as shown in the left panel of Figure 3. We note that only the density range of 8 ≤ log n(H) (${\mathrm{cm}}^{-3}$ ) ≤ 12 is considered, since below n(H) = 10⁸ ${\mathrm{cm}}^{-3}$ the clouds are inefficient in producing emission lines and above n(H) = 10¹² ${\mathrm{cm}}^{-3}$ the clouds mostly produce thermalized continuum emission rather than emission lines (Korista & Goad 2000). In addition, we only sum the contributions from ionized clouds with $6\leqslant \mathrm{log}\,{{\rm{U}}}_{{\rm{H}}}{\rm{c}}\leqslant 11.25$ (see details in Korista & Goad 2000; Korista et al. 2004; Korista & Goad 2019).

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In the left panel of Figure 3, we also consider the case where the continuum luminosity at 3000 Å of the quasar drops by a factor of 10, i.e., ${L}_{3000\mathring{\rm A} }$ decreases from 10⁴⁵ to 10⁴⁴ $\mathrm{erg}\ {{\rm{s}}}^{-1}$ (gray shaded area in Figure 4), corresponding to Q(H) ≃ [10^55.5, 10^54.5] (s⁻¹). When the quasar continuum changes from the bright state (solid lines) to the faint state (lighter lines) by 1 dex, the radial emissivity function simply shifts to the left by 0.5 dex (since ${\rm{\Phi }}({\rm{H}})\propto L/{r}^{2}$) assuming no dynamical structure changes of the BLR, as well as its inner and outer boundaries. Mg ii emissivity changes much less than Balmer lines in our assumed BLR, and this difference in the level of line emissivity changes from the bright state to the faint state increases with increasing radius.

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Following previous works (e.g., Goad et al. 1993; Korista et al. 2004), we define a responsivity parameter

$$\begin{eqnarray}&&\eta =\displaystyle \frac{d\mathrm{log}F({r})}{d\mathrm{log}{\rm{\Phi }}({\rm{H}})}\propto -0.5\displaystyle \frac{d\mathrm{log}F({r})}{d\mathrm{log}{r}},\,\,\mathrm{since}\ {\rm{\Phi }}({\rm{H}})\propto {{r}}^{-2},\end{eqnarray} \tag{ 6 }$$

to describe the efficiency of converting the change in the ionizing continuum flux to the change in the responding line flux. Integrating over the full line-emitting region, we have the relation ${L}_{\mathrm{line}}\propto {L}_{\mathrm{con}}^{\eta }$. Furthermore, if we assume $F(r)\propto {r}^{\gamma }$ with γ = −1, then η = −γ/2 = 0.5, which is a rough approximation for several UV/optical emission lines (e.g., Goad et al. 2012; see also the dashed gray lines in Figure 3).

The right panel of Figure 3 shows that the responsivities η of Balmer lines and Mg ii are correlated with the emitting radius and anticorrelated with the incident luminosity L(t) at different times. Since the clouds with $\eta (r,L({\rm{t}}))\gt 1$ and <0 will not respond appropriately to the variations in the ionizing continuum (Korista et al. 2004), we generally consider the 0 < η < 1 region. Since η starts to exceed 0 at R = 10^16.5 cm for both the bright and faint states, we adopt R_in = 10^16.5 cm (12 lt-day) for our LOC model. As long as the inner boundary is small, it has a minor impact on our results because the clouds there have very high density (>10¹² cm⁻³) and low responsivity η, with cloud emission dominated by continuum and not emission lines. However, the outer BLR boundary is important in our LOC model, which will determine the "breathing" of the broad emission lines (see Section 3.2). Note that although η will be larger than 1 at R_out = 10¹⁸ cm in the faint state, most of these clouds are emitting inefficiently and will have little impact on our results.

To calculate the line luminosity, we still need to specify the distribution functions of clouds, i.e., f(r) and g(n). Traditionally LOC models have used empirical parameterizations for these distribution functions aiming at reproducing the observed emission-line properties. For example, according to Baldwin et al. (1995), f(r) ∝ r^Γ and g(n) ∝ n^β (Γ = −1 and β = −1) are simple and reasonable assumptions for BLR clouds. This parameterization of f(r) and g(n) results in equal weighting for each grid point in the density–flux plane in log scale. For the best-known NGC 5548, Korista & Goad (2000) suggested that −1.4 < Γ < −1 is the optimized range to recover the observed time-averaged UV spectrum in 1993, and hence it was fixed to −1.2 in Korista & Goad (2019). In this work we fix Γ = −1.1 to match the Mg ii luminosity observed in a rare Mg ii changing-look quasar (CLQ; see the details in Section 4.2). Hence, Equation (5) becomes

$$\begin{eqnarray}&&{L}_{\mathrm{line}}\propto \int d(\mathrm{log}n){\int }_{{R}_{\mathrm{in}}}^{{R}_{\mathrm{out}}}{r}^{1.9}F(r)d(\mathrm{log}r).\end{eqnarray} \tag{ 7 }$$

Given the luminosity range of the quasar, we obtain the LOC predicted L_con–L_line relation in Figure 4. In the left panel, Mg ii varies at a slower rate with continuum than the Balmer lines, particularly in the gray region that encloses the typical luminosity range of a CLQ (e.g., MacLeod et al. 2016; Yang et al. 2018). In this shaded region, when the continuum luminosity drops by 1 dex, the Mg ii luminosity is only reduced by 0.45 dex, which is less than the luminosity reduction in Balmer lines (e.g., 0.6 dex for Hα and 0.7 dex for Hβ). In the right panel, we show different line ratios as a function of continuum luminosity. With decreasing central luminosity, the ratios of Hα/Hβ and Mg ii/Hβ increase, which is consistent with the theoretical results in Baldwin et al. (1995) and observations of the quasar composite spectrum (Vanden Berk et al. 2001) and CLQs (MacLeod et al. 2016, 2019; Yang et al. 2018).

Given the long dynamical timescale (more than a few decades) in the dust sublimation region, the physical outer boundary of the BLR is determined by the average luminosity state at least decades ago and could be largely unrelated to the current luminosity state. Here we explore the possibility that R_out deviates from that estimated based on the current luminosity and the consequences on the breathing behaviors of different broad lines.

If we consider that the previous L_{3000 Å} that set the outer BLR boundary is 10 times smaller or larger than the current average luminosity, we obtain R_out = 10^17.5 or 10^18.5 cm assuming $\mathrm{log}{\rm{\Phi }}({\rm{H}})=17.9\ {\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ and that the BLR outer boundary has not yet had enough time to dynamically adjust itself owing to luminosity state changes. Thus, with three values of ${R}_{\mathrm{out}}={10}^{17.5},{10}^{18}$, and 10^18.5, we produce three cases that have different behaviors in the relation between line width and luminosity (e.g., second and third rows of Figure 5): in the left column, Hβ shows "breathing," but Mg ii and Hα only cover half of the luminosity range (partially breathing); in the middle column, none of the lines show "breathing" (no-breathing); and in the right column, all three lines show "breathing" (see details below).

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The resulting radial distributions of line emission are presented in the top panels of Figure 5. Similarly, we consider a continuum change of 1 dex from the bright state (solid lines) to the faint state (lighter lines) to mimic a CLQ. These radial emissivity profiles will not only shift to the left but also move down owing to Φ(H) ∝ L/r². The Mg ii emissivity peak is always located at larger radii than those for the Balmer lines. Moreover, the Mg ii and Hα emissivity profiles are more narrowly distributed than that of Hβ. In the left column, the maximum emissivity for all three lines is located very close to the outer boundary of the BLR in the bright state. In the faint state, the maximum emissivity of Hβ is shifted to inside the outer boundary, while the maximum emissivities of Mg ii and Hα are still near the outer boundary.

Assuming that the BLR is virialized,¹⁰ we compute the average virial velocity and the corresponding observed broad-line width (line dispersion for both ${\sigma }_{\mathrm{peak}}$ and σ_eff) for each line and display the results in the bottom panels of Figure 5. The σ_peak is calculated based on the peak-emissivity radius R_peak. This simplification is adopted to demonstrate the concept of different "breathing" modes since the line emission is nearly dominated by the emissivity peak (Baldwin et al. 1995). We also consider a more realistic line width estimation, the effective line width σ_eff weighted by the radial line emissivity:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{eff}}=\displaystyle \frac{\int \sigma {r}^{1.9}F(r)d\,\mathrm{log}\,(r)}{\int {r}^{1.9}F(r)d\,\mathrm{log}\,(r)},\end{eqnarray} \tag{ 8 }$$

where $\sigma =\sqrt{\tfrac{{M}_{\mathrm{BH}}G}{{Rf}}}$, ${M}_{\mathrm{BH}}={10}^{8.5}{M}_{\odot }$, and f = 4.47.

As shown in Figure 3, the σ_peak-based cases clearly present distinctions between breathing and no-breathing regions, and hence we use it to define three breathing categories. However, for more realistic situations, all σ_eff-based cases show identical anticorrelations, but only with different slopes more skewing toward breathing or no-breathing scenarios. Both estimations demonstrate that Mg ii is always the least breathing among the three lines for R_out = 10¹⁸ cm. Comparing to the typical observed line width (σ ∼ 1700 $\mathrm{km}\,{{\rm{s}}}^{-1}$) for Hβ in SDSS quasars with ${M}_{\mathrm{BH}}={10}^{8.5}{M}_{\odot }$, σ_eff (∼1600 $\mathrm{km}\,{{\rm{s}}}^{-1}$) is preferred over the use of σ_peak (∼1000 $\mathrm{km}\,{{\rm{s}}}^{-1}$). In addition, since the left column is the most consistent with observed properties for the Balmer lines and Mg ii (e.g., Park et al. 2012; Shen 2013; Yang et al. 2019), we will use the σ_eff-based partially breathing model as our fiducial model to produce further relations and CL sequence in Sections 3.3 and 4.2. This model is the same fiducial LOC model for all other predictions.

The intrinsic Baldwin effect (BE; Baldwin 1977) states that the line EW decreases with increasing continuum luminosity for a given quasar. From Equation (6),

$$\begin{eqnarray}&&\eta =\displaystyle \frac{d\mathrm{log}F({r})}{d\mathrm{log}{\rm{\Phi }}({\rm{H}})}=\displaystyle \frac{d\mathrm{log}{\rm{EW}}}{d\mathrm{log}{\rm{\Phi }}({\rm{H}})}+1,\end{eqnarray} \tag{ 9 }$$

and therefore ${L}_{\mathrm{con}}\propto {{EW}}^{\eta -1}$, which means that the slope of the intrinsic BE is governed by the responsivity η that varies with the continuum luminosity and the formation radius. For instance, the responsivity η for Hβ in partially breathing BLRs for the bright (faint) state is ∼0.5 (0.7) at the most effective formation radius R_eff = 10^17.52 (10^17.25) cm (see Figures 4 and 9). Thus, the BE slope varies between −0.5 and −0.3 from the bright to the faint states, which is consistent with the prediction (∼−0.4) in Figure 6. For the well-studied AGN NGC 5548, the intrinsic BE slope ∼−0.6 predicted by LOC models has been confirmed by observations (e.g., Gilbert & Peterson 2003; Rakić et al. 2017), verifying the reliability of the LOC model. In addition, the average EW values of different emission lines (i.e., ${\mathrm{logEW}}_{\mathrm{MgII}}=1.5$ and ${\mathrm{logEW}}_{{\rm{H}}\beta }=1.8$ ) in Figure 6 are also consistent with observed values for SDSS quasars around ${L}_{3000\mathring{\rm A} }={10}^{44.5}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$ (Shen et al. 2011). Note that the EWs for Hβ and Hα are computed using ${L}_{5100\mathring{\rm A} }$, scaled from ${L}_{3000\mathring{\rm A} }$ by a factor of 1.8 based on the average quasar SED (Richards et al. 2006).

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On the other hand, the model also predicts a correlation between the line width and the EW. This correlation is driven by the combined effect of breathing and the BE: both line width and EW are anticorrelated with continuum luminosity. However, if Mg ii is partially breathing in the luminosity range probed in Figure 6, it predicts a steeper slope of ∼4 for Mg ii in logarithm space than the observed slope (∼1) in population studies (Dong et al. 2009; Shen et al. 2011). It is likely that the relations revealed here for a single quasar (i.e., the relation is intrinsic) contribute to the global relation observed for populations of quasars with different BH masses and luminosities.

As discussed in Section 1, Mg ii is an important broad line of RM interest at intermediate redshifts with optical spectroscopy. Confirming earlier photoionization calculations (e.g., Korista & Goad 2000; Korista et al. 2004), we found that the variability properties of broad Mg ii can be reasonably well explained by the LOC model. First, the broad Mg ii responds to the continuum variability at a lower level than the broad Balmer lines, which means that the Mg ii lags will be more difficult to detect in general. The fact that the Mg ii gas on average is located at slightly larger distance than the Balmer line gas also adds to the difficulty of detecting Mg ii lags, since longer monitoring duration is required to capture the lag. Moreover, there is a general mismatch between the formation radius of the lines and the characteristic timescale of the driving continuum, which means that in general the measured delays, and thus inferred "sizes," are underestimated (Perez et al. 1992; Goad & Korista 2014).

Perhaps a more striking feature of Mg ii variability is the general lack of breathing when luminosity changes. We have argued that this lack of breathing could be explained by the possibility that the Mg ii gas with maximum emission efficiency is near the outer physical boundary of the BLR, and hence the flux-weighted radius of the Mg ii-emitting clouds does not vary much with luminosity.¹¹ Alternatively, if the inner and outer boundaries of the BLR are fixed and η(r, L(t)) is constant over radius, which means that the responsivity η is the same for the line core and wings, the line will not display breathing, as the entire profile scales up and down with continuum variation (e.g., Korista & Goad 2004). A third possibility is that if the outer part of the BLR is dominated by turbulent motion (or nonvirial motion), the line width of Mg ii will also be less sensitive to continuum variations (e.g., Goad et al. 2012). The lack of breathing for Mg ii suggests that there is no intrinsic R−L relation for the Mg ii BLR. However, a global R−L relation may still exist for quasars over a broad BH mass and luminosity range, if the outer radius R_out scales with BH mass. Figure 7 demonstrates the possible existence of a global R−L relation and the absence of an intrinsic R−L relation for Mg ii. More RM results on Mg ii will be important to test the existence, or lack thereof, of a global R−L relation (e.g., Shen et al. 2015, 2016; Czerny et al. 2019).

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One assumption in our photoionization modeling is that the physical BLR structure does not change over the period of the continuum variability. It is possible that R_out will slowly change on dynamical timescale at this radius (>10² yr for our default parameters of M_BH = 10^8.5 M_⊙ and R_out ∼ 0.3 pc) and may eventually settle down at a different value in a new average luminosity state over the much longer lifetime of the quasar. This possibility could also be responsible for a global R−L relation for Mg ii.

Our LOC models also have implications for the behaviors of broad-line responses to continuum changes in the population of optically identified CLQs, or more generally, extreme variability (or hypervariable) quasars (e.g., Rumbaugh et al. 2018).

Most of the CLQs reported so far are spectroscopically defined by dramatic changes in the broad Balmer line flux between the bright and the dim states (e.g., LaMassa et al. 2015; MacLeod et al. 2016, 2019; Runnoe et al. 2016; Sheng et al. 2017; Yang et al. 2018). In most, if not all, of these cases, broad Mg ii remains visible even in the dim state (e.g., MacLeod et al. 2019; Yang et al. 2019).

With our LOC models, we can qualitatively explain the persistence of broad Mg ii in a CL event. Figure 8 presents a time sequence of synthetic spectra with the continuum luminosity decreasing from 10⁴⁵ to ${10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, for the fiducial model (R_out = 10¹⁸ cm) that can reproduce the observed Mg ii variability properties (see Sections 3.1 and 3.2). Each spectrum consists of a quasar power-law continuum, a host galaxy component,¹² and broad/narrow emission lines (e.g., Hα, Hβ, Mg ii, [O iii], and [S ii]) described by single-Gaussian functions. In the faintest state we only include the host galaxy and narrow emission lines. We adopt a quasar continuum power-law slope α = −1.56 (f_λ = λ^α; e.g., Vanden Berk et al. 2001). We start at the brightest epoch with ${L}_{3000\mathring{\rm A} }={10}^{45}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$ and reduce the continuum luminosity in steps of 0.15 dex. The strengths of the broad emission lines are computed from our fiducial LOC model at each continuum luminosity. Note that the index Γ for the radial distribution of cloud coverage determines the line luminosity and the BH mass governs the line dispersion (since the effective radius is determined from photoionization). For our fiducial LOC model, the M_BH is fixed to ${10}^{8.5}{M}_{\odot }$, which yields σ_eff = 10^3.4 $\mathrm{km}\,{{\rm{s}}}^{-1}$ at continuum luminosity of 10⁴³ $\mathrm{erg}\ {{\rm{s}}}^{-1}$ . Meanwhile, Mg ii luminosity is required to be ∼10^41.5 $\mathrm{erg}\ {{\rm{s}}}^{-1}$ at this continuum luminosity to match the only reported Mg ii CLQ in Guo et al. (2019), which led to the choice of Γ = −1.1 in Section 2 for our fiducial LOC model.

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From the sequence in Figure 8, it is obvious that when the broad Balmer lines almost disappear (e.g., become undetectable), the broad Mg ii emission remains visible. When the continuum luminosity continues to drop, broad Mg ii will eventually become too weak to be detectable. Indeed, we have found an example of an Mg ii CL object from a systematic search with repeated SDSS spectra (Guo et al. 2019). In that case, the Mg ii equivalent width dramatically changed from ∼100 Å to being consistent with zero, with little continuum change due to the dominance of the host light at the dim state. In Figure 9, we calculate the ${\sigma }_{\mathrm{eff}}$-based average radius for each epoch in this CL sequence, which decreases when luminosity decreases. Most importantly, the average radius (line width) of Mg ii decreases (increases) more slowly than those of the Balmer lines, verifying our fiducial model in Figures 5 and 6.

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We note that there is currently some ambiguity in the observational definition of "CLQs" based on the appearances of the broad Balmer lines, i.e., the detection of broad Balmer lines is dependent on signal-to-noise ratio. Our LOC calculations demonstrate that the variability of the broad emission lines in a CLQ can be fully explained by photoionization responses to the dramatic continuum changes. There is nothing special about the properties of the broad emission lines in CLQs compared to normal quasars, except for their extreme continuum variability. For these reasons, "extreme variability quasars," or "hypervariable quasars," is a more appropriate category term for these objects in our opinion.

Given the sequence of broad-line spectra shown in Figure 8 following the fading of continuum emission, it is possible to catch the quasar in a state where there is detectable broad Mg ii but no detectable broad Balmer lines. Roig et al. (2014) discovered that there is a rare population of broad Mg ii emitters in spectroscopically confirmed massive galaxies from the SDSS. We postulate that these broad Mg ii emitters may be the transition quasar population where the quasar continuum and broad Balmer line flux had recently dropped by a large factor but the broad Mg ii flux is still detectable on top of the stellar continuum. Using the sample of broad Mg ii emitters from Roig et al. (2014), we confirmed that the EWs of the broad Mg ii follow the extrapolated Baldwin effect in our LOC model to the lower luminosity range¹³ of ${L}_{3000}\sim {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Thus, the LOC model provides a natural explanation for these rare broad Mg ii emitters in otherwise normal galaxy spectra.

NGC 5548 is the ideal case to compare with our LOC model predictions given its extensive optical/UV RM data in the past several decades. We collected the published RM results from the 13 yr AGN Watch project (Peterson et al. 2004) and the six-month Space Telescope and Optical Reverberation Mapping (AGN STORM) project (Pei et al. 2017) to investigate the relations between continuum and emission-line fluxes (i.e., Hβ). Unfortunately, for NGC 5548 the data on Mg ii are much less than the data on Hβ, and therefore we do not consider Mg ii here.

As shown in Figure 10, the continuum flux (host corrected) at 5100 Å and line flux are well correlated over several decades (see also Goad & Korista 2014). The time lags (or emitting size) between the continuum and Hβ display the expected breathing during the AGN Watch program (e.g., Gilbert & Peterson 2003; Goad et al. 2004; Cackett & Horne 2006). At the end of AGN Watch around 2000, NGC 5548 was in a historic low state for some considerable time before returning to the average historic luminosity in the more recent AGN STORM campaign. If the outer BLR boundary was set during the low state around 2000, we would expect the 2014 AGN STORM RM results to have weaker Hβ strength and a reduced lag (since the BLR had been physically truncated), consistent with the results around 2000 (from AGN Watch) and in 2014 (STORM). Therefore, it is plausible that there is a physical outer boundary of the BLR set by the prior average luminosity state, as postulated in our model.

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