REVERBERATION MAPPING OF THE BROAD-LINE REGION IN NGC 5548: EVIDENCE FOR RADIATION PRESSURE?

The spectroscopic and photometric observations of NGC 5548 were made using the Yunnan Faint Object Spectrograph and Camera (YFOSC), mounted on the Lijiang 2.4 m telescope at Yunnan Observatory, Chinese Academy of Sciences. Working at the Cassegrain focus, YFOSC is a versatile instrument for low-resolution spectroscopy and photometry. It is equipped with a back-illuminated 2048 × 2048 pixel CCD, with pixel size 13.5 μm, pixel scale $0\buildrel{\prime\prime}\over{.} 283$ per pixel, and field of view $10^{\prime} \times 10^{\prime}$. YFOSC can automatically switch from spectroscopy to photometry within 1 s (see Du et al. 2014).

Our observations started on 2015 January 7, and ended on 2015 July 11. To get an accurate flux calibration, we simultaneously observed a nearby comparison star along the long slit as a reference standard. Such an observation strategy was described in detail by Maoz et al. (1990) and Kaspi et al. (2000), and was recently adopted by Du et al. (2014). As shown in Figure 1, we chose the star labeled "No.1" to be the comparison star. Given the seeing of $1\buildrel{\prime\prime}\over{.} 0\mbox{--}2\buildrel{\prime\prime}\over{.} 5$ throughout the monitoring period, we fixed the projected slit width at $2\buildrel{\prime\prime}\over{.} 5$. We used Grism 14, which provides a resolution of 92 Å mm⁻¹ (1.8 Å pixel⁻¹) and covers the wavelength range 3800−7200 Å. Standard neon and helium lamps were used for wavelength calibration. To reduce atmospheric differential refraction, we limited the observations to air masses $\lesssim 1.2$. This guarantees an atmospheric refraction $\leqslant 0\buildrel{\prime\prime}\over{.} 3$ over the wavelength range 4500–5500 Å (Filippenko 1982). The mean air mass for all spectra is 1.07, so that any offset of the target from the slit center due to atmospheric refraction is limited to $\lesssim 0\buildrel{\prime\prime}\over{.} 14\mbox{--}0\buildrel{\prime\prime}\over{.} 21$, which has a negligible impact on our analysis.

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To verify the calibration of the spectroscopic data, we also made photometric observations using a Johnson V filter. We took three consecutive exposures of 90 s each. In total, we obtained 62 spectroscopic observations and 61 photometric observations, spanning a time period of 180 days. The typical cadence is ∼3.4 days.

The two-dimensional spectroscopic images were reduced using the standard IRAF tools before absolute flux calibration. This includes bias subtraction, flat-field correction, and wavelength calibration. All the spectra were extracted using a uniform aperture of 30 pixels (85), and the background was determined from two adjacent regions on either side of the aperture region. As described in Du et al. (2014), absolute flux calibration was done in two steps. (1) The observations taken during nights with good weather conditions were used to calibrate the absolute flux of the comparison star, which was then used as the fiducial spectrum for absolute flux standard for the science observations. (2) For each object/comparison star pair, a wavelength-dependent sensitivity function was obtained by comparing the star's spectrum to the fiducial spectrum. This sensitivity function was then applied to calibrate the observed spectrum of the target. The spectra calibrated in this way show a small fluctuation of the [O iii] $\lambda 5007$ flux at a level of 2%, which can be regarded as the accuracy of our absolute flux calibration.

The V-band images were also reduced using standard IRAF (V2.16) procedures. Instrumental magnitudes were measured with respect to five selected reference stars in the field (see Figure 1). The typical accuracy of the photometry is 0.015 mag.

Following Hu et al. (2015), we measured the 5100 Å continuum and broad Hβ line using a spectral fitting scheme. The fitting components include (1) a single power-law continuum, (2) a stellar component for the host galaxy, (3) Fe  ii emission, (4) broad Hβ emission line, (5) broad He  ii $\lambda 4686$ emission line, and (6) narrow emission lines of [O iii] $\lambda \lambda 4959,5007$, He  ii $\lambda 4686$, Hβ, and several coronal lines (such as [Fe vii] $\lambda 5158$, [Fe vi] $\lambda 5176$ and [Ca v] $\lambda 5309;$ see details in Hu et al. 2015). The spectral template for the host galaxy is a stellar population model with an age of 11 Gyr and a metallicity Z = 0.05 (Bruzual & Charlot 2003).

During the observations, seeing variations and miscentering of the object in the slit led to varying amounts of host galaxy light in the final spectrum. To remove this effect, the flux of the host galaxy component was set to a free parameter in our fitting scheme. During the whole monitoring period, the ${\rm{H}}\beta$ emission line of NGC 5548 showed extreme broad wings, in addition to a strong narrow component. We slightly modified the fitting scheme of Hu et al. (2015) in two aspects, by adding a narrow component to ${\rm{H}}\beta$ and using three Gaussians to fit the broad component. Figure 2 shows the fitting results for the mean spectrum of our campaign and for an individual spectrum.

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We measured the 5100 Å continuum from the best-fit power-law component and the ${\rm{H}}\beta$ flux by integrating the best-fit broad components of ${\rm{H}}\beta$ from 4710 to 5050 Å. To construct the velocity-resolved delay map (see Section 3.3), we obtained the broad ${\rm{H}}\beta$ profile by subtracting the host galaxy, Fe  ii and [O iii] emission lines, and the narrow component of Hβ. Table 1 summarizes the light curves. Figure 3 plots the light curves of the V-band photometry (instrumental magnitude in an arbitrary unit), 5100 Å continuum, and broad ${\rm{H}}\beta$ flux.

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Table 1.  Continuum and ${\rm{H}}\beta$ fluxes for NGC 5548

JD	${F}_{{\rm{5100}}}$	${F}_{{\rm{H}}\beta }$	JD	V-band
−2450000			−2450000	(mag)
7030.50	5.46 ± 0.29	8.00 ± 0.24	7030.50	1.38 ± 0.05
7037.50	4.96 ± 0.29	7.96 ± 0.24	7037.50	1.45 ± 0.01
7043.50	5.67 ± 0.33	7.89 ± 0.27	7044.50	1.45 ± 0.01
7047.50	4.65 ± 0.32	7.78 ± 0.26	7046.50	1.44 ± 0.01
7055.50	4.17 ± 0.28	7.84 ± 0.24	7047.50	1.46 ± 0.01

Note. ${F}_{{\rm{5100}}}$ is the flux density at 5100 Å in units of ${10}^{-15}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}\,{\mathring{\rm A} }^{-1}$ and ${F}_{{\rm{H}}\beta }$ is the Hβ flux in units of ${10}^{-13}\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Following standard practice (e.g., Rodríguez-Pascual et al. 1997), we calculate the variability amplitude of the light curves of the 5100 Å continuum and Hβ emission line using

$$\begin{eqnarray}&&{F}_{{\rm{var}}}=\displaystyle \frac{{({\sigma }^{2}-{{\rm{\Delta }}}^{2})}^{1/2}}{\langle F\rangle }\,\end{eqnarray} \tag{ 1 }$$

and its uncertainty (Edelson et al. 2002)

$$\begin{eqnarray}&&{\sigma }_{{}_{{F}_{{\rm{var}}}}}=\displaystyle \frac{1}{{F}_{{\rm{var}}}}{\left(\displaystyle \frac{1}{2N}\right)}^{1/2}\displaystyle \frac{{\sigma }^{2}}{\langle F\rangle },\end{eqnarray} \tag{ 2 }$$

where $\langle F\rangle ={N}^{-1}{\sum }_{i=1}^{N}{F}_{i}$ is the average flux, F_i is the flux of the ith observation of the light curve, N is the total number of observations, ${\sigma }^{2}={\sum }_{i=1}^{N}{({F}_{i}-\langle F\rangle )}^{2}/(N-1)$, ${{\rm{\Delta }}}^{2}\ ={\sum }_{i=1}^{N}{{\rm{\Delta }}}_{i}^{2}/N$, and ${{\rm{\Delta }}}_{i}$ is the uncertainty of F_i. Table 2 lists the statistics of the light curves. The variability amplitudes of the 5100 Å continuum and broad Hβ line are ${F}_{{\rm{var}}}=0.23$ and 0.10, respectively, which are generally comparable with those of previous RM campaigns, after correcting for host galaxy and narrow Hβ contributions (Peterson et al. 2002; Bentz et al. 2007; Denney et al. 2010). We also calculate another standard measure of variability, ${R}_{{\rm{\max }}}$, simply defined as the ratio of maximum to minimum flux. The values of ${R}_{{\rm{\max }}}$ are 2.31 and 1.52 for the 5100 Å continuum and broad Hβ line, respectively.

Table 2.  Statistics of Light Curves for NGC 5548 in 2015

Time Series	N	$\langle T\rangle$	${T}_{{\rm{median}}}$	Mean Fluxa	${F}_{{\rm{var}}}$	${R}_{{\rm{\max }}}$
		(days)	(days)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
5100 Å	62	3.4	3.0	4.34 ± 0.30	0.23 ± 0.02	2.31
${\rm{H}}\beta$	62	3.4	3.0	6.95 ± 0.25	0.10 ± 0.01	1.52

Note. Column (1) is time series. Column (2) is the number of data points. Columns (3) and (4) are the mean and median sampling intervals, respectively. Column (5) is the mean flux and standard deviation. Columns (6) and (7) are ${F}_{{\rm{var}}}$ and ${R}_{{\rm{\max }}}$, defined in Section 3.1.

^aThe units of F₅₁₀₀ and ${F}_{{\rm{H}}\beta }$ are the same as in Table 1.

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We compute the time lag of the Hβ line flux relative to the 5100 Å continuum using the interpolation cross-correlation function method (ICCF; Gaskell & Sparke 1986; Gaskell & Peterson 1987; White & Peterson 1994). The time lag is measured by two standard approaches: the peak location ${\tau }_{{\rm{peak}}}$ of the ICCF (${r}_{{\rm{\max }}}$) and the centroid ${\tau }_{{\rm{cent}}}$ of the ICCF around the peak above a typical value ($r\geqslant 0.8\,{r}_{{\rm{\max }}}$). Their respective uncertainties were obtained using the Monte Carlo "flux randomization/random subset sampling" method described by Peterson et al. (1998, 2004). The Monte Carlo simulations were run with 1000 realizations, and the distributions of the peak and centroid (CCPD and CCCD) were created from the generated samples. The uncertainties of ${\tau }_{{\rm{peak}}}$ and ${\tau }_{{\rm{cent}}}$ are then calculated from the CCPD and CCCD, respectively, with a 68.3% confidence level ($1\sigma$).

In Figures 3(d) and (e), we show the auto cross-correlation function (ACF) of the light curve of the 5100 Å continuum, the ICCF between the Hβ flux and 5100 Å continuum, and the CCPD and CCCD of the ICCF, respectively. We find that the ICCF peak occurs at ${\tau }_{{\rm{peak}}}={7.20}_{-0.35}^{+1.33}$ days (${r}_{{\rm{\max }}}=0.83$) and the ICCF centroid occurs at ${\tau }_{{\rm{cent}}}={7.18}_{-0.70}^{+1.38}$ days in the rest frame (see Table 3). As an independent check, we also calculate the Z-transformed discrete correlation function (ZDCF; Edelson & Krolik 1988; Alexander 1997) and superpose the corresponding ZDCF in Figures 3(d) and (e). As can be seen, the ZDCFs are in good agreement with the ICCFs.

Table 3.  RM Measurements

Parameter	Value
${\tau }_{{\rm{cent}}}$ (${\rm{H}}\beta$ versus F5100)	${7.20}_{-0.35}^{+1.33}$ days
${\tau }_{{\rm{peak}}}$ (${\rm{H}}\beta$ versus F5100)	${7.18}_{-0.70}^{+1.38}$ days
FWHM (rms)	9450 ± 290 km s−1
${\sigma }_{{\rm{line}}}(\mathrm{rms})$	3124 ± 302 km s−1
FWHM (mean)	9912 ± 362 km s−1
${\sigma }_{{\rm{line}}}(\mathrm{mean})$	3350 ± 272 km s−1
$\mathrm{log}({\bar{L}}_{{\rm{5100}}}/\mathrm{erg}\,{{\rm{s}}}^{-1}$)	43.21 ± 0.12
$\mathrm{log}({\bar{L}}_{{\rm{H}}\beta }/\mathrm{erg}\,{{\rm{s}}}^{-1}$)	41.70 ± 0.05

Note. ${\tau }_{{\rm{cent}}}$ and ${\tau }_{{\rm{peak}}}$ are given in the rest-frame.

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Velocity-resolved RM is widely used to reveal the kinematic signatures of BLRs (e.g., Bentz et al. 2008, 2010; Denney et al. 2009a, 2009b, 2010; Grier et al. 2013; De Rosa et al. 2015; Du et al. 2016b). The high-quality spectroscopic data of our campaign allow us to construct the velocity-binned delay map of the Hβ line. Our procedure is as follows. Using the method described in the Appendix, we first calculate the rms spectrum of the broad Hβ profiles obtained in Section 2.3. As illustrated in Figure 4(a), we then select a wavelength range from 4731 Å to 4991 Å in the rest frame and divide the ${\rm{H}}\beta$ profiles into nine uniformly spaced bins (each bin has a velocity width of $\sim 1700\,\ \mathrm{km}\,{{\rm{s}}}^{-1}$).⁸ The light curve of each bin is finally obtained by just integrating the flux in the bin. The time lag of each bin and the associated uncertainties are determined using the same procedures as described in Section 3.2.

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In Figure 5, we plot the obtained light curves of the nine bins, along with the light curve of the 5100 Å continuum for the sake of comparison. The corresponding ICCFs between the light curve of each bin and the continuum are shown in the right panels of Figure 5, in which the CCCD and CCPD are also superposed. The velocity-resolved delay map is plotted in Figure 4(b). We can find that the delay map has a symmetric pattern, with longer response at the line core and shorter response at the wings (except for bin 5).

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Velocity-resolved delay maps of the broad Hβ line in NGC 5548 were derived previously in the MDM campaign undertaken in 2007 (Denney et al. 2009a, 2009b) and in the LAMP campaign in 2008 (Bentz et al. 2009). The delay map of our campaign is very similar to that of the MDM campaign. Denney et al. (2009a) conclude that the symmetric delay map of NGC 5548 indicates that there is no radial gas motion in the BLR. The data quality of the LAMP campaign is relatively poor so that the resulting delay map in Bentz et al. (2009) does not reveal clear signatures for the BLR motion. However, Pancoast et al. (2014b) carried out dynamical modeling of the LAMP data and found that a narrow thick-disk-like BLR geometry with dominant inflows can explain the variations of the broad Hβ line. On the other hand, it is worth mentioning that the C iv velocity-resolved delay map of the AGN STORM campaign derived by De Rosa et al. (2015) shows a symmetric structure. In the future, detailed analysis of the present data through dynamical modeling (e.g., Pancoast et al. 2011; Li et al. 2013) will better constrain the kinematics of the Hβ BLR.

We employ the method of Peterson et al. (2004) to measure the FWHM and ${\sigma }_{{\rm{line}}}$ of ${\rm{H}}\beta$ from the mean and rms spectra (Table 3; Appendix). We find $\mathrm{FWHM}/{\sigma }_{{\rm{line}}}\approx 3$, which is larger than 2.35 for a Gaussian profile, indicating that the BLR has less turbulent motion (Kollatschny & Zetzl 2013). We calculate the virial product (Table 4)

$$\begin{eqnarray}&&{\rm{VP}}=\displaystyle \frac{c{\tau }_{{}_{{\rm{H}}\beta }}{V}^{2}}{G},\end{eqnarray} \tag{ 3 }$$

where ${\tau }_{{}_{{\rm{H}}\beta }}$ is the Hβ time lag, c is the speed of light, G is the gravitational constant, and V is the line width. If the BLR motion is dominated by the gravity of the central BH and is virialized, VP should be constant, and $V\propto {\tau }_{{}_{{\rm{H}}\beta }}^{-1/2}$.

Figure 6 shows the relation between V and ${\tau }_{{}_{{\rm{H}}\beta }}$ using four measures of line width, namely, ${\sigma }_{{\rm{line}}}$ and FWHM from the mean and rms spectra. The relations between V and ${\tau }_{{}_{{\rm{H}}\beta }}$ have a slope of (−0.54, −0.56, −0.56, −0.40) for ${\sigma }_{{\rm{line}}}$ and the FWHM of the rms and mean spectra, respectively. Generally, the Hβ line width obeys the virial relation within the uncertainties, consistent with the results reported by Peterson et al. (2004) and Bentz et al. (2007). To quantitatively describe any deviation of the BLR motion from the virial relation, we define a parameter

$$\begin{eqnarray}&&\delta =\displaystyle \sum _{i}^{N}{\delta }_{{V}_{i}}^{2},\,\,\,\text{and}{\delta }_{{V}_{i}}=\mathrm{log}\left(\displaystyle \frac{{V}_{i}}{{V}_{{\rm{vir}}}}\right),\end{eqnarray} \tag{ 4 }$$

where ${V}_{{\rm{vir}}}={(G\langle {\rm{VP}}\rangle /{R}_{{}_{{\rm{BLR}}}})}^{1/2}$ and $\langle {\rm{VP}}\rangle$ is the average virial product. We find that the deviation $\delta =(0.11,0.10,0.10,0.19)$ for ${\sigma }_{{\rm{line}}}$ and the FWHM of the rms and mean spectra, respectively.

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Figures 7(a) and (b) show the distribution of VP as a function of optical luminosity. Compared with the previous analysis of Collin et al. (2006) and Bentz et al. (2007), we extend the analysis by including the latest RM measurements. Using the procedure FITEXY of Press et al. (1992), we obtain the regressions

$$\begin{eqnarray}&&\mathrm{log}({\rm{VP}}/{M}_{\odot })\\ =\left\{\begin{array}{ll}(7.13\pm 0.04)+(0.21\pm 0.13)\mathrm{log}{{\ell }}_{43} & (\mathrm{for}\,{\sigma }_{{\rm{line}}}),\\ (7.97\pm 0.05)+(0.36\pm 0.20)\mathrm{log}{{\ell }}_{43} & (\text{for FWHM}),\end{array}\right.\end{eqnarray} \tag{ 5 }$$

where ${{\ell }}_{43}={\bar{L}}_{5100}/{10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Figures 7(c) and (d) show that the distributions of VP${| }_{{\sigma }_{{\rm{line}}}}$ and VP${| }_{{\rm{FWHM}}}$ have a large scatter (0.31 and 0.32 dex, respectively). We note that the weak correlation between VP and luminosity is not in conflict with the notion that the BLR is predominantly virialized. Even if variations in radiation pressure induces secular departures from virial equilibrium, the kinematics gradually adjusts to restore a quasi-virialized state. This test shows that the method for determining the BH mass is robust as long as the factor ${f}_{{}_{{\rm{BLR}}}}$ is reliably calibrated.

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Peterson et al. (2004) found that the ${R}_{{}_{{\rm{BLR}}}}\mbox{--}{\bar{L}}_{{\rm{5100}}}$ relation of NGC 5548 has a much steeper slope than 0.5. Figure 8(a) re-emphasizes this point by including the new measurement from our campaign. Using FITEXY, we obtain the best-fit regression of

$$\begin{eqnarray}&&\mathrm{log}({R}_{{\rm{H}}\beta }/{\rm{ltd}})=(0.94\pm 0.05)+(0.86\pm 0.18)\mathrm{log}{{\ell }}_{43},\end{eqnarray} \tag{ 6 }$$

where ${{\ell }}_{43}={\bar{L}}_{5100}/{10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. With the addition of the new data, our derived relation is slightly steeper than that reported by Kilerci Eser et al. (2015; ${R}_{{}_{{\rm{BLR}}}}\propto {L}_{5100}^{0.79\pm 0.2}$). Both slopes are steeper than the value of 0.5 expected from simple photoionization theory. For completeness, Figure 8(b) also shows the best-fit regression of the relation between the Hβ BLR size and the Hβ luminosity, ${R}_{{\rm{H}}\beta }\propto {\bar{L}}_{{\rm{H}}\beta }^{0.87\pm 0.25}$. Recently, based on data from simultaneous optical and UV observations, Kilerci Eser et al. (2015) established the connection between the 5100 Å and the 1350 Å luminosity as ${\bar{L}}_{{\rm{5100}}}\propto {\bar{L}}_{{\rm{1350}}}^{0.63\pm 0.12}$. Using this relation, we deduce ${R}_{{\rm{H}}\beta }\propto {\bar{L}}_{{\rm{1350}}}^{0.54\pm 0.22}$, which is consistent with the expected slope of 0.5. This nonlinear relation between the optical and UV emissions implies a complicated geometry for the accretion disk and the importance of radiative reprocessing (e.g., Edelson et al. 2015; Fausnaugh et al. 2015).

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Following standard practice, we estimate the BH mass as

$$\begin{eqnarray}&&{M}_{\bullet }={f}_{{}_{{\rm{BLR}}}}\displaystyle \frac{c{\tau }_{{}_{{\rm{H}}\beta }}{V}^{2}}{G},\end{eqnarray} \tag{ 7 }$$

where ${f}_{{}_{{\rm{BLR}}}}$ is a coefficient that crudely accounts for the unknown inclination, geometry, and kinematics of the BLR. Ho & Kim (2014) point out that the BLR in pseudobulges notably has a lower ${f}_{{\rm{BLR}}}$ than in classical bulges. For line dispersion ${\sigma }_{{\rm{line}}}$ measured from the rms spectra, ${f}_{{}_{{\rm{BLR}}}}=3.2\pm 0.7$ for pseudobulges, whereas ${f}_{{}_{{\rm{BLR}}}}=6.3\pm 1.5$ for classical bulges. The latter is roughly consistent, within uncertainties, with previous calibrations that do not take the bulge type into consideration (e.g., ${f}_{{}_{{\rm{BLR}}}}=5.5\pm 1.7;$ Onken et al. 2004). NGC 5548 hosts a classical bulge (Ho & Kim 2014). Using ${f}_{{}_{{\rm{BLR}}}}=6.3\pm 1.5$, we derive a BH mass of ${M}_{\bullet }{| }_{{\sigma }_{{\rm{line}}}}={8.71}_{-2.61}^{+3.21}\times {10}^{7}{M}_{\odot };$ line dispersion measured from mean spectra, ${f}_{{}_{{\rm{BLR}}}}=5.6\pm 1.3$ (Ho & Kim 2014) and ${M}_{\bullet }{| }_{{\sigma }_{{\rm{line}}}}={8.91}_{-2.67}^{+3.08}\times {10}^{7}{M}_{\odot }$. The two mass measurements are in good agreement. Meanwhile, our measurements are also consistent, within the uncertainties, with the overall values from previous RM campaigns. A notable exception is the work of Pancoast et al. (2014b); applying the BLR dynamical modeling developed by Pancoast et al. (2014a) to the LAMP data on NGC 5548, they obtain, without invoking the virial factor, ${M}_{\bullet }={3.89}_{-1.49}^{+2.87}\times {10}^{7}{M}_{\odot }$, which is only about half of the virial-based value.

The classical bulge of NGC 5548 has a central stellar velocity dispersion of ${\sigma }_{* }=195\pm 13$ km s⁻¹ (Woo et al. 2010), resulting in ${M}_{\bullet }{| }_{{\sigma }_{* }}=(2.75\pm 0.88)\times {10}^{8}{M}_{\odot }$ from the latest ${M}_{\bullet }\mbox{--}{\sigma }_{* }$ relation (Kormendy & Ho 2013). This is marginally larger than the virial-based BH mass estimate, after taking into account the intrinsic scatter of the virial factor (Ho & Kim 2014), but significantly exceeds the BH mass determination based on dynamical modeling of the BLR by Pancoast et al. (2014b).

With the BH mass in hand, we can calculate the dimensionless accretion rate, defined as

$$\begin{eqnarray}&&\dot{{\mathscr{M}}}=\displaystyle \frac{{\dot{M}}_{\bullet }{c}^{2}}{{L}_{{\rm{Edd}}}}\approx 0.1\,{\eta }_{0.1}^{-1}\left(\displaystyle \frac{{L}_{{\rm{bol}}}}{{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{M}_{\bullet }}{{10}^{8}{M}_{\odot }}\right)}^{-1},\end{eqnarray} \tag{ 8 }$$

where ${\dot{M}}_{\bullet }$ is the mass accretion rate, ${L}_{{\rm{Edd}}}=1.5\ \times {10}^{38}({M}_{\bullet }/{M}_{\odot })\mathrm{erg}\,{{\rm{s}}}^{-1}$ is the Eddington luminosity, ${L}_{{\rm{bol}}}\ =\eta {\dot{M}}_{\bullet }{c}^{2}$ is the bolometric luminosity (see Table 5), and ${\eta }_{0.1}=\eta /0.1$ is the radiative efficiency of the accretion disk. Using a BH mass of ${M}_{\bullet }=8.71\times {10}^{7}\,{M}_{\odot }$ and a mean bolometric luminosity of ${L}_{{\rm{bol}}}={10}^{44.33}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Table 5), we obtain $\dot{{\mathscr{M}}}=0.21$ for $\eta =0.1$. The BH in NGC 5548 has an accretion rate in the regime of the standard accretion disk model (Shakura & Sunyaev 1973). Note that the above accretion rate is inaccurate if NGC 5548 hosts a binary supermassive BH, as recently suggested by Li et al. (2016) based on the detection of periodic variations in luminosity and velocity.

Table 5.  Simultaneous Observations of the Spectral Energy Distribution for NGC 5548

References	Observations	$\mathrm{log}({L}_{{\rm{bol}}}/\mathrm{erg}\,{{\rm{s}}}^{-1})$	${\epsilon }_{{}_{{\rm{Edd}}}}$	Notes
V09a	XMM(OM, EPIC-pn) in 2000 Dec	44.35 ± 0.04	0.018 ± 0.002	simultaneous
V09b	Swift(UVOT, XRT) in 2007 Jul	44.15 ± 0.04	0.011 ± 0.002	simultaneous
V10	Swift(BAT) in 2007 Jul, IRAS in 1983	44.50 ± 0.04	0.025 ± 0.003	not simultaneous

References. V09a: Vasudevan & Fabian (2009a), V09b: Vasudevan et al. (2009b), V10: Vasudevan et al. (2010). There are many observations of NGC 5548 (see Chiang & Blaes 2003 for a brief summary of data). Here we only list the recent observations. "Simultaneous" means that the UV, optical, and X-ray data are simultaneously observed.

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Figure 9 shows the variations of the mean ${\bar{L}}_{5100}$ and ${R}_{{}_{{\rm{BLR}}}}$ over all the 18 campaigns since 1989. The variation amplitude of ${\bar{L}}_{5100}$ exceeds an order of magnitude. The lowest ${L}_{5100}\sim {10}^{42.2}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ occurred in 2005−2009 (Bentz et al. 2007, 2009; Denney et al. 2010) and the highest ${L}_{5100}\ \sim {10}^{43.5}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ in 1998–1999 (Peterson et al. 1999, 2002). Meanwhile, ${R}_{{}_{{\rm{BLR}}}}$ also exhibits large variations, from ∼5 up to ∼30 lt-days. Simple inspection of Figure 9 reveals that the changes in ${R}_{{}_{{\rm{BLR}}}}$ follow the variations of ${\bar{L}}_{5100}$, but plausibly with a time delay. Using the same procedure for computing Hβ lags, we determine the lag of ${R}_{{}_{{\rm{BLR}}}}$ with respect to ${\bar{L}}_{5100}$: ${\tau }_{{}_{R-\bar{L}}}\ ={2.35}_{-1.25}^{+3.47}$ years. From analysis of the CCCD and CCPD, the probability of ${\tau }_{{}_{R-\bar{L}}}\leqslant 0\,{\rm{years}}$ p = 0.06 and 0.098, respectively, suggesting that there may be a potential delay between ${R}_{{}_{{\rm{BLR}}}}$ and ${\bar{L}}_{5100}$. Clearly, we need more RM monitoring campaigns to verify this intriguing but tentative result.

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There are two timescales for the ionized clouds in the BLR. The recombination timescale is ${t}_{{\rm{rec}}}={({n}_{e}{\alpha }_{{\rm{B}}})}^{-1}\ =6.0\,{n}_{10}^{-1}$ minutes, where ${n}_{10}={n}_{e}/{10}^{10}\,{\mathrm{cm}}^{-3}$ is the electron density of the clouds and ${\alpha }_{{\rm{B}}}$ is the case B recombination coefficient (Osterbrock 1989). The fast response of the ionization front exactly follows the ionizing luminosity, as shown by Equation (6). However, radiation pressure, if effective, drives the cloudsoutwards, changes their orbits, and leads to variations of their spatial distribution. The observed size of the BLR is actually an emissivity-averaged value over the whole BLR, namely ${R}_{{}_{{\rm{BLR}}}}=\int R\epsilon {dR}/\int \epsilon {dR}$, where is the emissivity of the clouds, determined by their spatial distribution (or, equivalently, their number density). Radiation pressure induces changes in , and thereby ${R}_{{}_{{\rm{BLR}}}}$. Considering that the Hβ line width represents the bulk motion of the clouds, the dynamical timescale of the BLR is given by (Peterson 1993)

$$\begin{eqnarray}&&{t}_{{}_{{\rm{BLR}}}}=\displaystyle \frac{c{\tau }_{{}_{{\rm{H}}\beta }}}{{V}_{{\rm{FWHM}}}}=3.36\,{\tau }_{{}_{{\rm{20}}}}{V}_{{\rm{5000}}}^{-1}\,{\rm{years}},\end{eqnarray} \tag{ 9 }$$

where ${V}_{{\rm{5000}}}={V}_{{\rm{FWHM}}}/5000\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${\tau }_{{}_{{\rm{20}}}}={\tau }_{{}_{{\rm{H}}\beta }}/20\,{\rm{days}}$. The quantity ${t}_{{}_{{\rm{BLR}}}}$ represents the typical timescale with which ${R}_{{}_{{\rm{BLR}}}}$ varies in response to a change in the BLR dynamics. For NGC 5548, the average Hβ lag of ${\tau }_{{}_{{\rm{H}}\beta }}=15$ days and line width of ${V}_{{\rm{FWHM}}}$ = 6000 km s⁻¹ leads to a dynamical timescale of ${t}_{{}_{{\rm{BLR}}}}\approx 2.10\,{\rm{years}}$.

Interestingly, ${t}_{{}_{{\rm{BLR}}}}\approx {\tau }_{{}_{R-\bar{L}}}$, indicates that the BLR could be jointly controlled by radiation pressure and BH gravity. A mass inflow to the center can give rise to variations of the BLR and the accretion disk, but changes in the BLR structure (size) should precede changes in the disk luminosity. The delayed response of the BLR size to continuum luminosity may rule out this possibility, plausibly implicating the potential role of radiation pressure.