Reverberation Mapping of Optical Emission Lines in Five Active Galaxies

In spring of 2014 we monitored 11 AGNs over the course of a six-month RM campaign. The AGNs were selected with the aim of expanding the database of RM SMBH masses, particularly for objects with diverse and peculiar observational characteristics. The second goal of our campaign was to investigate the dynamics and geometry of the BLR with velocity-resolved reverberation signatures, i.e., velocity-delay maps and dynamical models (see, e.g., Grier et al. 2013b; Pancoast et al. 2014a). Here, we focus on results related to SMBH masses and we will pursue the velocity-resolved analysis in future work.

Figure 1 shows g-band light curves from the Las Cumbres Observatory (LCO) 1 m network for nine of our targets (we discuss these data in detail in Section 2.3). Not shown are Akn 120, which was dropped early in the campaign because of low variability, and NGC 5548, for which the results are presented elsewhere (Fausnaugh et al. 2016; Pei et al. 2017). In order to estimate a black hole mass, we must measure a continuum-line lag. We have not been able to measure such a reverberation signal for Mrk 668, NGC 3227, CBS 0074, and PG 1244+026. These sources have lower signal-to-noise ratios (S/Ns) than the other objects (generally 30–70 per pixel, although NGC 3227 was ∼90 per pixel; see Section 2.5.3), and they display lower variability amplitudes. The fractional root-mean-square amplitude (F_var as defined in Section 2.5.3 below) is 0.012 for Mrk 668, 0.037 for NGC 3227, 0.010 for CBS 0074, and 0.025 for PG 1244+026. For Mrk 668, the slow rate of change in the light curve also makes it impossible to measure short lags. For NGC 3227, the light curve is problematic because of the limited sampling and large gaps; however, this object was also observed during a monitoring campaign in 2012, and we will combine the data from both campaigns in a future analysis. For CBS 0074 and PG 1244+026, we have not been able to obtain a sufficiently precise calibration of the spectra (see Section 2.2.2) to detect emission-line variability.

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We succeeded in measuring black hole masses for MCG+08-11-011, NGC 2617, NGC 4051, 3C 382, and Mrk 374. Table 1 lists the some of the important properties of these objects (several of which are measured in this study), and we provide additional comments as follows.

• i.  
  MCG+08-11-011 is a strong X-ray source for which spectral signatures of a relativistically broadened Fe Kα line have been observed with Suzaku (Bianchi et al. 2010). The Fe Kα emission is believed to be emitted close to the inner edge of the accretion disk and can potentially be used to measure the spin parameter of the black hole. Because the black hole mass and spin are to some extent degenerate when fitting the broad Fe Kα profile, an independent mass estimate from RM can greatly assist with the spin measurement.
• ii.  
  NGC 2617 was discovered by Shappee et al. (2014) to be a "changing look" AGN. In 2013, after a large X-ray/optical outburst, follow-up spectroscopic observations showed the presence of broad lines, while archival spectra from 2003 show only a weak broad component of Hα. This means that the classification of NGC 2617 changed from a Seyfert 1.9 to Seyfert 1.0 sometime in the intervening decade. Few optical "changing look" AGNs are known, although systematic searches through long-term survey data (such as the SDSS, LaMassa et al. 2015; MacLeod et al. 2016) and targeted repeat spectroscopy (Ruan et al. 2016; Runco et al. 2016; Runnoe et al. 2016) have recently expanded the sample size to approximately 20 objects, depending on how "changing look" AGNs are defined. The absolute rate of this phenomenon is very uncertain, but these recent studies suggest that it may be relatively common over several decades, a timescale that long-term spectroscopic surveys are only beginning to probe. Velocity-resolved dynamical information is of special interest in an object such as this, since the presence of outflows or infall may provide clues about the physical mechanism behind the change in Seyfert category.
• iii.  
  NGC 4051 has been the target of several optical and X-ray RM campaigns (Peterson et al. 2000, 2004; Shemmer et al. 2003; Denney et al. 2009b; Miller et al. 2010; Turner et al. 2016). However, the short Hβ lag, comparable to the cadence of most monitoring campaigns, has led to mixed and inconsistent results. Denney et al. (2009b) found an Hβ lag of 1.87 ± 0.52 days, roughly a factor of two smaller than previous studies. Because of the large change, as well as the lag's small value compared to the monitoring cadence, we re-observed NGC 4051 during the 2014 campaign to check this result. For one month of the campaign (2014 February 17 to March 16 UTC), we also increased the monitoring cadence of NGC 4051 to twice nightly, in order to securely resolve the expected short Hβ lag.NGC 4051 is also an archetypal narrow-line Seyfert 1 (NLS1), meaning that the width of its Hβ line is ≲2000 km s⁻¹. There are two competing theories to explain the NSL1 phenomenon: high accretion rates or rotationally dominated BLR dynamics seen nearly face-on. Both explanations can account for the narrow linewidths given the AGN luminosity. Insight into the structure of the BLR can help distinguish between these explanations, so there is considerable interest in reconstructing a velocity-delay map for this object.
• iv.  
  3C 382 is an FR II broad-line radio galaxy (Osterbrock et al. 1975, 1976). Few radio-loud AGNs have RM mass measurements, although there are notable examples such as 3C 390 (Shapovalova et al. 2010; Dietrich et al. 2012), 3C 273 (Kaspi et al. 2000; Peterson et al. 2004), and 3C 120 (Peterson et al. 2004; Grier et al. 2012). These objects are typically more luminous than radio-quiet AGNs, so they have large lags (on the order months to years) that are difficult and expensive to measure. However, radio emission is thought to be associated with more massive black holes, which can be tested by anchoring radio-loud AGNs to the RM mass scale. Radio jets can also provide an indirect estimate of the inclination of the BLR, if the BLR is a disky structure with the rotation axis aligned to that of the jet (Wills & Browne 1986). Several jet-orientation indicators exist for 3C 382, and Eracleous et al. (1995) estimated the BLR inclination in 3C 382 using dynamical models of the double-peaked Hα profile. Velocity-delay maps and dynamical models would provide an interesting comparison to these estimates.
• v.  
  We observed Mrk 374 in an RM campaign from 2012, but this AGN did not display sufficient variability to measure emission-line lags at that time. Although Mrk 374 is our least variable source, we succeeded in measuring a line lag from the 2014 campaign, and we present the first RM-based black hole mass here.

Table 1.  Target Properties

Object	Redshift	DL	Number of	F [O iii]λ5007	[O iii]λ5007	$\mathrm{log}\lambda {L}_{5100\mathring{\rm{A}} }$	$\mathrm{log}\lambda {L}_{\mathrm{host}}$	E(B−V)
		(Mpc)	Good-weather Epochs	(10−14 erg cm−2 s−1)	Light Curve Scatter (%)	(erg s−1)	(erg s−1)	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MCG+08-11-011	0.0205	89.1	6	61.33 ± 0.21	0.09	43.59	43.28	0.19
NGC 2617	0.0142	61.5	3	6.88 ± 0.06	1.37	43.12	42.95	0.03
NGC 4051	0.0023	17.1	3	41.30 ± 0.26	0.36	42.38	42.23	0.01
3C 382	0.0579	258.7	6	7.87 ± 0.06	0.92	44.20	43.98	0.06
Mrk 374	0.0426	188.5	3	7.27 ± 0.11	0.62	43.98	43.61	0.05

Note. Column 2 is taken from the NASA Extragalactic Database. Column 3 gives the luminosity distance D_L in a concensus cosmology, except for NGC 4051 for which the luminosity distance is from Tully et al. (2008). Column 4 gives the number of nights with clear and stable conditions on which each object was observed. Each object had three observations per night, which were used to calculate the narrow [O iii]λ5007 line flux. The line flux and its uncertainty are given in Column 5. Column 6 gives the fractional variation of the [O iii]λ5007 line light curve, which serves as an estimate of the night-to-night calibration error (Section 2.5.1). Column 7 gives the observed luminosity (corrected for Galactic extinction), calculated from the observed 5100 Å rest-frame light curve and Column 3. Column 8 gives the luminosity of the host-galaxy starlight in the spectroscopic extraction aperture, also corrected for Galactic extinction (Section 5.1). Note that Column 7 includes the contribution from the host galaxy. Column 9 gives the Galactic reddening value from Schlafly & Finkbeiner (2011).

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2.2.1. Observations

2.2.2. Night-to-night Flux Calibration

Our spectroscopic observations are supplemented with broadband imaging observations. Contributing telescopes were the 0.7 m at the Crimean Astrophysical Observatory (CrAO), the 0.5 m Centurian 18 at Wise Observatory (WC18, Brosch et al. 2008), and the 0.9 m at West Mountain Observatory (WMO). CrAO uses an AP7p CCD with a pixel scale of 176 and a 15' × 15' field of view, WC18 uses a STL-6303E CCD with a pixel scale of 147 and a 75' × 50' field of view, and WMO uses a Finger Lakes PL-3041-UV CCD with a pixel scale of 061 and a field of view of 21' × 21'. Fountainwood Observatory (FWO) also provided observations of NGC 4051 with a 0.4 m telescope using an SBIG 8300M CCD. The pixel scale of this detector is 035 and the field of view is 19' × 15'. All observations were taken with the Bessell V-band.

In addition, we obtained ugriz imaging with the LCO 1 m network (Brown et al. 2013), which consists of nine identical 1 m telescopes at four observatories spread around the globe. These data were originally acquired as part of LCO's AGN Key project (Valenti et al. 2015). The main goal is to search for continuum reverberation signals, which we will pursue in a separate study (Fausnaugh et al. 2017). However, 3C 382 and Mrk 374, which are our faintest sources, had low variability amplitudes and poorer S/Ns, so we included the LCO g-band data in the continuum light curves of these objects. Each LCO telescope has the same optic system and detectors—at the time of the RM campaign, the detectors were SBIGSTX-16803 cameras with a field of view of 16' × 16' and a pixel scale of 023.

We analyzed the imaging data using the image subtraction software (ISIS) developed by Alard & Lupton (1998). Images were first uploaded to a central repository and vetted by eye for obvious reduction errors or poor observing conditions. We then registered the images to a common coordinate system and constructed a high S/N reference frame by combining the best-seeing and lowest-background images. When combining, ISIS adjusts the images to a common seeing by convolving the point-spread function (PSF) of each image with a spatially variable kernel. Finally, we subtracted the reference frame from each image, again allowing ISIS to match the PSFs using its convolution routine. Reference images and subtractions for each telescope/filter/detection system were constructed separately—we discuss combining the photometric measurements in Section 2.5.2.

Figures 2–6 show the noise-weighted mean spectrum

$$\begin{eqnarray}&&\overline{F}(\lambda )=\displaystyle \frac{{\sum }_{i=1}^{{N}_{t}}F(\lambda ,{t}_{i})/{\sigma }^{2}(\lambda ,{t}_{i})}{{\sum }_{i=1}^{{N}_{t}}1/{\sigma }^{2}(\lambda ,{t}_{i})}\end{eqnarray} \tag{ 1 }$$

for each object using the MDM observations, where F(λ, t_i) is the flux density at epoch t_i and σ(λ, t_i) is its uncertainty. Figures 2–6 also show root-mean-square (rms) residual spectra, defined as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{rms}}(\lambda )=\sqrt{\displaystyle \frac{1}{{N}_{t}-1}{\sum }_{i=1}^{{N}_{t}}{[F(\lambda ,{t}_{i})-\overline{F}(\lambda )]}^{2}}.\end{eqnarray} \tag{ 2 }$$

By the Wiener–Khinchin theorem, this statistic is proportional to the integrated variability power at each wavelength, so σ_rms is free of constant contaminants such as host-galaxy and narrow emission-line flux. However, the total variability power contains contributions from both intrinsic variations and from statistical fluctuations/measurement uncertainties. In order to separate these components, we use a maximum-likelihood method (see Park et al. 2012b; Barth et al. 2015; De Rosa et al. 2015). We solve for the intrinsic variability σ_var(λ) that minimizes the negative log-likelihood

$$\begin{eqnarray}-2\mathrm{ln}{ \mathcal L } & = & \displaystyle \sum _{i=1}^{{N}_{t}}\displaystyle \frac{{[F(\lambda ,{t}_{i})-\hat{F}(\lambda )]}^{2}}{{\sigma }^{2}(\lambda ,{t}_{i})+{\sigma }_{\mathrm{var}}^{2}(\lambda )}\\ & & +\,\displaystyle \sum _{i=1}^{{N}_{t}}\mathrm{ln}[{\sigma }^{2}(\lambda ,{t}_{i})+{\sigma }_{\mathrm{var}}^{2}(\lambda )],\end{eqnarray} \tag{ 3 }$$

where $\hat{F}(\lambda )$ is the "optimal average" weighted by ${\sigma }^{2}({t}_{i})+{\sigma }_{\mathrm{var}}^{2}$. We self-consistently fit for $\hat{F}(\lambda )$ while solving for σ_var(λ), and we show the estimate of σ_var(λ) with the red lines in Figures 2–6. In the limit that $\sigma (\lambda ,{t}_{i})\to 0$, it is clear that σ_var is equivalent to σ_rms. For high S/N data such as these, σ_var(λ) is nearly equal to ${[{\sigma }_{\mathrm{rms}}^{2}(\lambda )-{\overline{\sigma }}^{2}(\lambda )]}^{1/2}$, where ${\overline{\sigma }}^{2}(\lambda )$ is the average of the squared measurement uncertainties across the time series:

$$\begin{eqnarray}&&{\overline{\sigma }}^{2}(\lambda )=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{{N}_{t}}{\sigma }^{2}(\lambda ,{t}_{i}).\end{eqnarray} \tag{ 4 }$$

The overall effect is to reduce the squared amplitude of the variability spectrum by the mean squared measurement uncertainty—in all objects except for Mrk 374, this effect is negligible.

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2.5.1. Spectroscopic Light Curves

We extracted spectroscopic light curves for the wavelength windows listed in Table 2 for each AGN. We chose these windows based on visual inspection of the variable line profiles in the σ_var(λ) spectra, with the main goal of capturing the strongest variations in the lines. For 3C 382, the component tentatively identified as He ii λ4686 is blueshifted by almost 100 Å relative to the systematic redshift, and if variable He ii λ4686 has a similar profile as the Balmer lines in this object, this component corresponds to the blue wing of the line.

The rest-frame 5100 Å continuum, which is relatively free of emission/absorption lines, was estimated by averaging the flux density in the listed wavelength region. Emission-line fluxes were determined in the same way as for the [O iii] λ5007 line. First, we subtracted a linear least-squares fit to the local continuum underneath the emission line. Wavelength regions for the continuum fits are given in Table 3. Then, we integrated the remaining flux using Simpson's method (we did not assume a functional form for the emission line). In cases where the broad Hβ wing extends underneath [O iii]λ4959, we subtracted the narrow emission line (again with a local linear approximation of the underlying flux) before integrating the broad line. We did not attempt to separate the narrow components of Hβ and Hγ from the broad components. These narrow components act as constant flux-offsets for the light curves.

The continuum estimates can lead to significant systematic uncertainties because the continuum-fitting windows may be contaminated by broad-line wing emission, and the local linearly interpolated continuum may leave residual continuum flux to be included in the line profile. Both of these effects can introduce spurious correlations between the continuum and line light curves, which may biased the final lag estimates. Because we use the σ_var(λ) spectra to select the line and continuum windows, variability in the line wings probably does not have a large impact on our results, and we have found the the resulting light curves (and their lags) are robust to 5 to 10 angstrom changes in the continuum and line windows. Larger shifts, especially as the continuum-fitting windows move further from the lines, can result in significantly different lags (on the order of three times the statistical uncertainties). Full spectral decompositions may be able to address this issue in future studies (see Barth et al. 2015 for a detailed discussion). We discuss these systematic uncertainties further in Section 4.

After we extracted line fluxes from the WIRO and MDM spectra, we combined the measurements by forcing the light curves to be on the same flux scale. We used the mean MDM [O iii] λ5007 line to define this scale and multiplied the WIRO line fluxes so that the mean value matched that of MDM. A more sophisticated inter-calibration model would include an additive offset, to account for different amounts of host-galaxy starlight in the MDM and WIRO spectra. However, with the limited amount of WIRO data, additional calibration parameters cannot be well-constrained, and we found the simple multiplicative approach to be adequate. The required rescaling factors were 1.21 for MCG+08-11-011, 1.14 for NGC 4051, 1.09 for 3C 382, and 1.73 for Mrk 374. Weather at WIRO prevented observations of NGC 2617.

The statistical uncertainty on the continuum flux was estimated from the standard deviation within the wavelength region,

$$\begin{eqnarray}&&\sigma ({t}_{j})=\sqrt{\displaystyle \frac{1}{{N}_{\lambda }-1}{\sum }_{i=1}^{{N}_{\lambda }}{[F({\lambda }_{i},{t}_{j})-\overline{F}({t}_{j})]}^{2}},\end{eqnarray} \tag{ 5 }$$

where $\overline{F}({t}_{j})$ is the evenly weighted average flux density at epoch t_j. Uncertainties on the line light curves were estimated using a Monte Carlo approach: we perturbed the observed spectrum with random deviates scaled to the uncertainty at each wavelength, subtracted a new estimate of the underlying continuum (and the narrow [O iii] λ4959 line when appropriate), and re-integrated the line flux. The deviates were drawn from the multivariate normal distribution defined by the covariance matrix of the rescaled spectrum—these covariances can affect the statistical uncertainty by a factor of two or more (see Fausnaugh 2017 for more details). We repeated this procedure 10³ times and took the central 68% confidence interval of the output flux distributions as an estimate of the statistical uncertainty.

Because the integrated [O iii] λ5007 line flux is not explicitly forced to be equal from night to night, the scatter of the [O iii] λ5007 line light curve serves as an estimate of our calibration uncertainty (Barth et al. 2015). We extracted narrow [O iii] λ5007 line light curves in the same way as for the broad lines, and the results are shown in Figure 7. Several points are noticeably below the means of their light curves, particularly for NGC 2617 and 3C 382. These observations were taken in poor weather and display significant scatter between the individual rescaled exposures prior to averaging. This suggests variable amounts of flux-losses between the AGN and extended [O iii] λ5007/host galaxy, due to variable seeing and large guiding errors that move the object in the slit. Although the rescaling model from Section 2.2.2 cannot correct this issue, the offsets of these points are not very large compared to the statistical uncertainties (no more than 3.1σ), and we opt to include them in the analysis. Since the effect due to spatially extended [O iii] λ5007 emission is relatively small even in very poor conditions, it will be unimportant in good conditions.

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The fractional standard deviations of the narrow-line light curves are given in Table 1 and range between 0.1% and 1.4%. These values only represent our ability to correct for extrinsic variations (such as weather conditions) in the observed spectra. Additional systematic uncertainties dominate the epoch-to-epoch uncertainties of the light curves, including (but not limited to) the nightly sensitivity functions, continuum subtraction, and additional spectral components such as Fe ii emission. The latter two issues are especially problematic for the He ii λ4686 light curves.

To account for these systematics, we rescaled the light-curve uncertainties so that they approximate the observed flux variations from night to night. We selected three adjacent points, $F({t}_{j-1})$, F(t_j), and $F({t}_{j+1})$, linearly interpolated between $F({t}_{j-1})$ and $F({t}_{j+1})$, and measure ${\rm{\Delta }}=[F({t}_{j})-I({t}_{j})]/\sigma ({t}_{j})$, where I(t_j) is the interpolated value at t_j and σ(t_j) is the statistical uncertainty on F(t_j). The deviate Δ therefore measures the departure of the light curve from a simple linear model. We calculated Δ for j = 2 to ${N}_{t}-1$ (i.e., ignoring the first and last points), and we multiplied the statistical uncertainties σ(t_j) by the mean absolute deviation (MAD) $\overline{| {\rm{\Delta }}| }$. We also imposed a a minimum value of 1.0 on these rescaling factors. Inspection of the distribution of Δ shows that the residuals are reasonably (but not perfectly) represented by a Gaussian with a similar MAD value. This method ensures that the uncertainties account for any systematics that the rescaling model cannot capture. We have ignored the uncertainty in the interpolation I(t_j), so our method slightly overestimates the required rescaling factors. Monte Carlo simulations may be able to assess the importance of uncertainty in I(t_j) for future work. The rescaling factors are given in Table 4 and are fairly small, generally running between 1.0 and 2.0, with a mean of 1.8 and a maximum of 3.42 for the Hβ light curve in NGC 4051. NGC 4051 has the largest rescaling factors overall, which may be due to real short timescale variability that departs from our simple linear model (Denney et al. 2010). We therefore also experimented with using the unscaled light-curve uncertainties in our time-series analysis (Section 3) for this object. We found that our results do not sensitively depend on the scale of the uncertainties, although our Bayesian lag analysis (Section 3.2) indicates that the unscaled uncertainties are probably underestimated.

Table 4.  Light-curve Properties

Object	Light curve	Nobs	Δtmed	Uncertainty	$\bar{F}$	$\langle {\rm{S}}/{\rm{N}}\rangle$	Fvar	${({\rm{S}}/{\rm{N}})}_{\mathrm{var}}$	rmax
			(days)	Rescaling Factor
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
MCG+08-11-011	5100 Å	190	0.59	1.51	4.49	71.2	0.10	67.4	≡1
	Hβ	86	1.01	1.58	3.79	103.1	0.07	47.8	0.90 ± 0.01
	Hγ	82	1.01	1.52	1.29	34.2	0.09	19.2	0.84 ± 0.03
	He iiλ4686	86	1.01	1.46	0.31	6.8	0.44	19.6	0.78 ± 0.04
NGC 2617	5100 Å	161	0.92	1.81	5.17	57.2	0.09	44.4	≡1
	Hβ	61	1.01	1.91	3.31	39.5	0.10	21.1	0.61 ± 0.07
	Hγ	61	1.01	1.65	1.18	11.0	0.20	12.3	0.62 ± 0.07
	He iiλ4686	61	1.01	1.18	0.15	3.2	0.61	10.9	0.49 ± 0.08
NGC 4051	5100 Å	270	0.47	1.00	12.90	191.7	0.02	49.7	≡1
	Hβ	107	0.96	3.42	3.14	45.7	0.09	31.1	0.59 ± 0.05
	Hγ	98	0.99	2.48	1.92	26.1	0.08	13.7	0.50 ± 0.06
	He iiλ4686	107	0.96	3.25	1.59	12.9	0.11	10.2	0.47 ± 0.07
3C 382	5100 Å	209	0.56	1.17	3.18	148.5	0.09	131.3	≡1
	Hβ	81	1.00	1.70	2.06	43.4	0.05	14.5	0.83 ± 0.25
	Hγ	81	1.00	1.66	0.60	8.1	0.07	3.6	0.59 ± 0.24
	He iiλ4686	81	1.00	1.90	0.13	3.2	0.19	3.9	0.61 ± 0.47
Mrk 374	5100 Å	180	0.59	2.39	3.80	94.5	0.03	25.5	≡1
	Hβ	67	1.01	1.54	1.29	38.4	0.05	11.2	0.50 ± 0.07
	Hγ	67	1.01	1.79	0.56	18.7	0.04	3.8	0.42 ± 0.07
	He iiλ4686	67	1.01	1.41	0.18	8.2	0.25	11.9	0.56 ± 0.06

Note. Column 3 gives the number of observations in each light curve. Column 4 gives the median cadence. Column 5 gives the rescaling factor by which the statistical uncertainties are multiplied to account for additional systematic errors (see Section 2.5.1). Column 6 gives the mean flux level of each light curve. The rest-frame 5100 Å continuum light curves are in units of 10⁻¹⁵ erg cm⁻² s⁻¹ Å⁻¹, and the emission-line light curves are in units of 10⁻¹³ erg cm⁻² s⁻¹. Column 7 gives the mean signal-to-noise ratio $\langle {\rm{S}}/{\rm{N}}\rangle$. Column 8 gives the rms fractional variability defined in Equation (6). Column 9 gives the approximate S/N at which we detect variability (see Section 2.5.3). Column 10 gives the maximum value of the interpolated cross-correlation function (see Section 3.1).

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2.5.2.  Broadband Light Curves

2.5.3. Light-curve Properties

The final light curves are shown in Figures 2–6 and given in Tables 5–14. We characterize the statistical properties of the light curves in Table 4, reporting the median cadence, mean flux level, and average S/N. We also measure the light-curve variability using a technique similar to our treatment of the variability spectra σ_var(λ). In the presence of noise, it is necessary to separate the intrinsic variability from that due to measurement errors. We therefore define the intrinsic variability of the light curves as σ_var and solve for it by minimizing

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }=\displaystyle \sum _{i}^{{N}_{t}}\displaystyle \frac{{[F({t}_{i})-\hat{F}]}^{2}}{{\sigma }^{2}({t}_{i})+{\sigma }_{\mathrm{var}}^{2}}+\displaystyle \sum _{i}^{{N}_{t}}\mathrm{ln}[{\sigma }^{2}({t}_{i})+{\sigma }_{\mathrm{var}}^{2}],\end{eqnarray} \tag{ 6 }$$

where F(t_i) is the flux at epoch i, σ(t_i) is its uncertainty, and $\hat{F}$ is the optimal average flux (weighted by ${\sigma }^{2}({t}_{i})+{\sigma }_{\mathrm{var}}^{2}$). For small measurement uncertainties, the fractional variability ${\sigma }_{\mathrm{var}}/\hat{F}$ converges to the standard definition of the "excess variance" (Rodríguez-Pascual et al. 1997)

$$\begin{eqnarray}&&{F}_{\mathrm{var}}=\displaystyle \frac{1}{\overline{F}}\sqrt{\displaystyle \frac{1}{N-1}{\sum }_{i}^{N}{[F({t}_{i})-\overline{F}]}^{2}-{\overline{\sigma }}^{2}},\end{eqnarray} \tag{ 7 }$$

where $\overline{\sigma }$ is the time-averaged measurement uncertainty of the light curve. We therefore define ${F}_{\mathrm{var}}={\sigma }_{\mathrm{var}}/\hat{F}$ and report these values in Table 4. These values are slightly underestimated, since $\hat{F}$ is not corrected for constant components (such as host-galaxy starlight or narrow-line emission). We also approximate the S/N of the variability as

$$\begin{eqnarray}&&{({\rm{S}}/{\rm{N}})}_{\mathrm{var}}=\displaystyle \frac{{\sigma }_{\mathrm{var}}}{\overline{\sigma }\sqrt{2/{N}_{\mathrm{obs}}}}.\end{eqnarray} \tag{ 8 }$$

The $\sqrt{2/{N}_{\mathrm{obs}}}$ term enters because the variance of $\overline{\sigma }$ is expected to approximately scale as that of a reduced χ² distribution. However, this calculation assumes uncorrelated uncertainties, and a full analysis requires treatment of the red-noise properties of the light curve (see Vaughan et al. 2003).

Table 5.  MCG+08-11-011 Continuum Light Curve

HJD	Fλ	Telescope ID
(days)	(10−15 ergs s−1 cm−2 Å−1)
(1)	(2)	(3)
6639.5218	3.6279 ± 0.0448	CrAO
6649.4750	3.6413 ± 0.0845	CrAO
6653.4973	3.8005 ± 0.0631	CrAO
6656.3942	3.9176 ± 0.0425	CrAO
6661.6986	3.9866 ± 0.0764	MDM
6662.2042	3.9837 ± 0.0412	WC18

Note. Column 1 gives HJD−2,450,000 at mid-exposure. Column 2 give the continuum flux density and uncertainty. Column 3 identifies the contributing telescope (see Section 2.3).

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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Table 6.  NGC 2617 Continuum Light Curve

HJD	${F}_{\lambda }$	Telescope ID
(days)	(10−15 ergs s−1 cm−2 Å−1)
(1)	(2)	(3)
6639.6747	4.6776 ± 0.1089	CrAO
6643.6320	4.9717 ± 0.1357	CrAO
6644.5145	5.0946 ± 0.0871	CrAO
6646.0981	5.0904 ± 0.0892	WC18
6646.5287	5.4647 ± 0.0885	CrAO
6647.0990	5.3930 ± 0.1254	WC18

Note. Columns are the same as in Table 5.

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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Table 7.  NGC 4051 Continuum Light Curve

HJD	Fλ	Telescope ID
(days)	(10−15 ergs s−1 cm−2 Å−1)
(1)	(2)	(3)
6645.6113	12.7318 ± 0.0624	WC18
6646.6098	12.8314 ± 0.0668	WC18
6647.6275	13.0803 ± 0.0683	WC18
6648.5919	12.9598 ± 0.0499	WC18
6650.5178	12.8375 ± 0.0550	WC18
6653.5959	13.0814 ± 0.0956	WC18

Note. Columns are the same as in Table 5.

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Table 8.  3C 382 Continuum Light Curve

HJD	${F}_{\lambda }$	Telescope ID
(days)	(10−15 ergs s−1 cm−2 Å−1)
(1)	(2)	(3)
6670.6411	3.2490 ± 0.1390	WC18
6689.0237	3.0685 ± 0.0094	LCOGT1
6690.0279	3.0018 ± 0.0105	LCOGT1
6691.6169	3.0619 ± 0.1197	WC18
6693.0025	2.9948 ± 0.0106	LCOGT1
6696.9897	2.8652 ± 0.0094	LCOGT1

Note. Columns are the same as in Table 5.

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Table 9.  Mrk 374 Continuum Light Curve

HJD	Fλ	Telescope ID
(days)	(10−15 ergs s−1 cm−2 Å−1)
(1)	(2)	(3)
6663.7432	3.7706 ± 0.1030	MDM
6664.7221	3.7776 ± 0.0649	MDM
6665.5588	3.8676 ± 0.1114	WC18
6666.7292	3.7496 ± 0.0791	MDM
6667.7164	3.7902 ± 0.0850	MDM
6668.7316	3.8045 ± 0.1097	MDM

Note. Columns are the same as in Table 5.

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Table 10.  MCG+08-11-011 Line Light Curves

HJD	Hβ	Hγ	He ii	Telescope ID
(days)	(10−13 erg s−1 cm−2)
(1)	(2)	(3)	(4)	(5)
6661.6986	3.4660 ± 0.0288	1.2156 ± 0.0298	0.1923 ± 0.0416	MDM
6663.6924	3.4891 ± 0.0302	1.1841 ± 0.0291	0.2341 ± 0.0401	MDM
6664.6734	3.4978 ± 0.0346	1.1986 ± 0.0326	0.2352 ± 0.0407	MDM
6666.6277	3.4723 ± 0.0387	1.0566 ± 0.0388	0.3973 ± 0.0505	MDM
6667.6147	3.5429 ± 0.0365	1.1637 ± 0.0354	0.4047 ± 0.0471	MDM
6668.6285	3.5031 ± 0.0307	1.2122 ± 0.0333	0.3580 ± 0.0437	MDM

Note. Column 1 gives HJD–2,450,000 at mid-exposure. Columns 2–4 give the line fluxes and their uncertainties. Column 5 identifies the contributing telescope (see Section 2.3).

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Table 11.  NGC 2617 Line Light Curves

HJD	Hβ	Hγ	He ii	Telescope ID
(days)	(10−13 erg s−1 cm−2)
(1)	(2)	(3)	(4)	(5)
6661.8270	3.4179 ± 0.1151	1.2480 ± 0.1362	0.0439 ± 0.0614	MDM
6662.8456	3.4662 ± 0.0848	1.2049 ± 0.1039	0.0145 ± 0.0470	MDM
6664.8143	3.2528 ± 0.0756	1.0938 ± 0.0892	0.0779 ± 0.0370	MDM
6666.8417	3.3192 ± 0.0635	1.1110 ± 0.0744	0.1912 ± 0.0346	MDM
6667.8446	3.3112 ± 0.0621	1.2235 ± 0.0793	0.1744 ± 0.0363	MDM
6668.8507	3.2390 ± 0.0690	1.3410 ± 0.0837	0.1203 ± 0.0344	MDM

Note. Columns are the same as in Table 10.

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Table 12.  NGC 4051 Line Light Curves

HJD	Hβ	Hγ	He ii	Telescope ID
(days)	(10−13 erg s−1 cm−2)
(1)	(2)	(3)	(4)	(5)
6664.9552	2.9038 ± 0.0705	1.8998 ± 0.0729	1.3355 ± 0.1207	MDM
6666.8941	2.7895 ± 0.0616	1.8579 ± 0.0659	1.4384 ± 0.1032	MDM
6667.8962	2.7808 ± 0.0565	1.7772 ± 0.0605	1.3202 ± 0.0981	MDM
6668.9146	2.8279 ± 0.0708	1.8106 ± 0.0767	1.6972 ± 0.1251	MDM
6669.9111	2.8152 ± 0.0586	1.7299 ± 0.0599	1.5860 ± 0.1034	MDM
6670.9090	2.8217 ± 0.0657	1.8284 ± 0.0694	1.5306 ± 0.1166	MDM

Note. Columns are the same as in Table 10.

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Table 13.  3C 382 Line Light Curves

HJD	Hβ	Hγ	He ii	Telescope ID
(days)	(10−13 erg s−1 cm−2)
(1)	(2)	(3)	(4)	(5)
6739.9650	2.0891 ± 0.0519	0.6969 ± 0.0820	0.1308 ± 0.0429	MDM
6747.9747	2.0787 ± 0.0581	0.6508 ± 0.0864	0.1038 ± 0.0452	MDM
6748.9640	2.0885 ± 0.0447	0.6067 ± 0.0715	0.0901 ± 0.0388	MDM
6749.9571	2.0675 ± 0.0487	0.6094 ± 0.0785	0.0745 ± 0.0392	MDM
6751.9471	2.0665 ± 0.0520	0.6116 ± 0.0784	0.1327 ± 0.0392	MDM
6752.9466	2.0513 ± 0.0550	0.4939 ± 0.0833	0.1497 ± 0.0464	MDM

Note. Columns are the same as in Table 10.

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Table 14.  Mrk 374 Line Light Curves

HJD	Hβ	Hγ	He ii	Telescope ID
(days)	(10−13 erg s−1 cm−2)
(1)	(2)	(3)	(4)	(5)
6663.7432	1.2948 ± 0.0280	0.5368 ± 0.0232	0.2157 ± 0.0174	MDM
6664.7221	1.2944 ± 0.0274	0.5467 ± 0.0238	0.1965 ± 0.0170	MDM
6666.7292	1.3247 ± 0.0282	0.6074 ± 0.0243	0.1911 ± 0.0188	MDM
6667.7164	1.3559 ± 0.0291	0.5865 ± 0.0245	0.1980 ± 0.0193	MDM
6668.7316	1.3172 ± 0.0278	0.5855 ± 0.0245	0.2111 ± 0.0184	MDM
6669.7373	1.3207 ± 0.0282	0.5365 ± 0.0254	0.2015 ± 0.0181	MDM

Note. Columns are the same as in Table 5.

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With the exception of the 3C 382 Hγ, the 3C 382 He ii λ4686, and the Mrk 374 Hγ light curves, we detect variability in all of the other emission lines at greater than ∼10σ. The variability amplitudes of MCG+08-11-011 and NGC 2617 are especially strong (F_var ≳ 10%). For NGC 4051, the continuum has little fractional variability (F_var = 2%), which may be caused by a high fraction of host-galaxy starlight. For MCG+08-11-011, NGC 2617, and NGC 4051, the median cadence is near 1 day for all light curves, and the mean S/N usually ranges from several tens to hundreds. In fact, the S/N in the spectra is even higher, reaching 100 to 300 per pixel in the continuum. Combined with the large variability amplitudes, it is likely that we will be able to construct velocity-delay maps and dynamical models for these objects in future work.

The cross-correlation procedure derives a lag from the centroid of the interpolated cross-correlation function (ICCF, Gaskell & Peterson 1987), as implemented by Peterson et al. (2004). For a given time delay, we shift the abscissas of the first light curve, linearly interpolate the second light curve to the new time coordinates, and calculate the correlation coefficient r_cc between all overlapping data points. We then repeat this calculation but shift the second light curve by the negative of the given time delay and interpolate the first light curve. The two values of r_cc are averaged together and the ICCF is evaluated by repeating this procedure on a grid of time delays spaced by 0.1 days. All ICCFs are measured relative to the 5100 Å continuum light curve (inter-calibrated with the broadband measurements). For each line light curve, the maximum value r_max of the ICCF is given in Table 4. The lag is estimated with the ICCF centroid, defined as ${\tau }_{\mathrm{cent}}=\int \tau {r}_{{cc}}(\tau )d\tau /\int {r}_{{cc}}(\tau )d\tau$ for values of ${r}_{{cc}}\geqslant 0.8{r}_{\max }.$

We estimate the uncertainty on τ_cent using the flux randomization/random subset sampling (FR/RSS) method of Peterson et al. (2004). This technique generates perturbed light curves by randomly selecting (with replacement) a subset of the data from both light curves and adjusting the fluxes by a Gaussian deviate scaled to the measurement uncertainties. The lag τ_cent is calculated for 10³ perturbations of the data and its uncertainty is estimated from the central 68% confidence interval of the resulting distribution. The ICCF and centroid distributions are shown in Figure 8 for all objects and line light curves, and Table 15 gives the median values and central 68% confidence intervals of these distributions. For completeness, we also report in Table 15 the lag τ_peak that corresponds to r_max. Note that these lags have been corrected to the rest frame of the source. For 3C 382, we do not find meaningful centroids in the ICCFs of the Hγ and He ii λ4686 light curves. This is because of the width of the autocorrelation function of the continuum and its poor correlation with the line light curves. We therefore do not include these lines for the rest of the ICCF analysis.

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Table 15.  Rest-frame Line Lags

Object	Line	τcent	τpeak	${\tau }_{{\mathtt{JAV}}}$	τmulti
		(days)	(days)	(days)	(days)
(1)	(2)	(3)	(4)	(5)	(6)
MCG+08-11-011	Hβ	${15.72}_{-0.52}^{+0.50}$	${15.02}_{-1.08}^{+1.86}$	${15.06}_{-0.28}^{+0.26}$	${14.98}_{-0.28}^{+0.34}$
	Hγ	${13.14}_{-1.05}^{+1.12}$	${12.08}_{-0.98}^{+0.69}$	${11.92}_{-0.44}^{+0.44}$	${12.38}_{-0.49}^{+0.46}$
	He iiλ4686	${1.88}_{-0.64}^{+0.58}$	${1.59}_{-0.98}^{+1.27}$	${1.24}_{-0.29}^{+0.36}$	${1.21}_{-0.33}^{+0.29}$
NGC 2617	Hβ	${4.32}_{-1.35}^{+1.10}$	${4.16}_{-1.68}^{+1.08}$	${6.22}_{-0.54}^{+0.51}$	${6.38}_{-0.50}^{+0.44}$
	Hγ	${0.91}_{-1.08}^{+1.50}$	${0.61}_{-0.89}^{+1.28}$	${0.83}_{-0.59}^{+0.58}$	${0.81}_{-0.61}^{+0.59}$
	He iiλ4686	${1.59}_{-0.69}^{+0.49}$	${1.79}_{-0.89}^{+0.20}$	${1.78}_{-0.35}^{+0.30}$	${1.75}_{-0.38}^{+0.34}$
NGC 4051	Hβ	${2.87}_{-1.33}^{+0.86}$	${2.42}_{-1.90}^{+1.00}$	${2.41}_{-0.46}^{+0.38}$	${2.24}_{-0.28}^{+0.39}$
	Hγ	${2.82}_{-2.84}^{+1.82}$	${2.82}_{-3.09}^{+1.90}$	${4.87}_{-0.08}^{+0.28}$	${2.40}_{-0.73}^{+0.86}$
	He iiλ4686	${0.27}_{-0.40}^{+0.33}$	${0.23}_{-0.40}^{+0.40}$	${0.06}_{-0.17}^{+0.17}$	$-{0.03}_{-0.15}^{+0.16}$
3C 382	Hβ	${40.49}_{-3.74}^{+8.02}$	${43.58}_{-3.50}^{+4.16}$	${52.07}_{-9.46}^{+3.18}$	⋯
Mrk 374	Hβ	${14.84}_{-3.30}^{+5.76}$	${14.81}_{-3.55}^{+5.85}$	${15.03}_{-1.26}^{+1.41}$	${13.73}_{-1.02}^{+1.06}$
	Hγ	${12.31}_{-9.80}^{+9.82}$	${12.51}_{-11.80}^{+9.69}$	${15.44}_{-2.85}^{+3.26}$	${13.37}_{-2.08}^{+2.11}$
	He iiλ4686	$-{1.53}_{-5.79}^{+3.21}$	$-{1.59}_{-5.85}^{+2.69}$	$-{0.44}_{-0.68}^{+0.71}$	$-{0.57}_{-0.64}^{+0.65}$

Note. Columns 3 and 4 give the centroids and peaks, respectively, of the interpolated cross-correlation functions (ICCFs). The uncertainties give the central 68% confidence intervals of the ICCF distributions from the FR/RSS procedure (see Section 3.1). Column 5 gives the lag fit by JAVELIN. Column 6 gives the same but using all light curves from a single object simultaneously. The uncertainties give the central 68% confidence intervals of the JAVELIN posterior lag distributions. All lags are relative to the 5100 Å continuum light curve and corrected to the rest frame. The uncertainties only represent the statistical errors—choices of continuum windows, detrending procedures, etc., introduce additional systematic uncertainties.

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Long-term trends in the light curves can bias the resulting ICCF due to red-noise leakage (Welsh 1999). We therefore experimented with detrending the light curves and/or restricting the baseline over which to calculate the ICCF. For MCG+08-11-011, these experiments had no effect, while for Mrk 374 and 3C 382 they eliminated any lag signal in the data. For NGC 2617, we found that restricting the data to 6620 < HJD − 2,450,000 < 6730 improved the ICCF by narrowing the central peak, as shown in the top four panels of Figure 9. However, this restriction changed the ICCF centroid by only 0.01 days, a negligible amount. For NGC 2617, the peaks in the Hγ and He ii λ4686 ICCFs at ±25 days are also obvious aliases, so we only report the lag based on the peak near 0 days. For NGC 4051, we found that detrending the continuum and line light curves with a second-order polynomial improves the ICCF, as shown in the bottom four panels of Figure 9. The long-term continuum trend is very weak, but there is a strong positive trend in the line light curves that is dominated by the linear term. Subtracting this linear trend decreases the median of the centroid distribution from 4.92 to 2.56 days, a change of 1.5σ. We adopt the smaller lag because of the quality of the detrended ICCF, and our Bayesian method (described below) finds a lag consistent with this smaller value.

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We also investigated the line lags using a Bayesian approach, as implemented by the JAVELIN software (Zu et al. 2011). JAVELIN explicitly models the reverberating light curves and corresponding transfer functions so as to find a posterior probability distribution of lags. We have already discussed JAVELIN's assumption that light curves are reasonably characterized by a DRW (Section 2.5.2). JAVELIN also assumes that the transfer function is a simple top-hat that can be parameterized by a width, an amplitude, and a mean time delay. This assumption is not very restrictive, since it is difficult to distinguish among transfer functions in the presence of noise (Rybicki & Kleyna 1994; Zu et al. 2011) and a top-hat is broadly consistent with expectations for physically plausible BLR geometries (e.g., disks or spherical shells).

We ran JAVELIN models for each line using the 5100 Å continuum as the driving light curve, and we used internal JAVELIN routines to remove any linear trends from the light curves during the fit. The damping timescale (a parameter of the DRW model) for most AGNs is several hundred days or longer (Kelly et al. 2009; MacLeod et al. 2010), and our light curves are not long enough to meaningfully constrain this parameter. We therefore (arbitrarily) fixed the damping timescale to 200 days. We also tested several different damping timescales (from a few days to 500 days), and found that the choice of 200 days does not affect the best-fit lags—an exact estimate of the damping timescale is not necessary to reasonably interpolate the light curves (Kozłowski 2016b). Table 15 gives the median and 68% confidence interval of the posterior lag distributions, denoted as ${\tau }_{{\mathtt{JAV}}}$. We also employed models that fit all light curves from a single object simultaneously, which maximizes the available information. These results are given in Table 15 as τ_multi. Posterior distributions of τ_multi are shown by the blue histograms in Figure 8. For the Hγ and He ii λ4686 light curves from 3C 382, we were again unable to constrain any lag signal, and we drop these light curves from the rest of this analysis.

We generally find consistent results between the ICCF method and JAVELIN models. The largest discrepancies are the Hβ lags for NGC 2617 (Δτ = 1.6σ) and 3C 382 (Δτ = 2.0σ), but these differences are not statistically significant. In NGC 2617, where the ICCF method detects a lag consistent with zero in the Hγ or He ii λ4686 light curves, JAVELIN finds a lag at reasonably high confidence: the percentiles for τ_multi = 0 in the posterior lag distributions of Hγ and He ii λ4686 are 8.3% and 1.1%, which are 1.4σ and 2.3σ detections for Gaussian probability distributions, respectively. For Mrk 374, a Hγ lag is detected at high significance using JAVELIN (we do not claim a lag detection for He ii λ4686 in this object, since the τ_multi = 0 percentile is 20%, only 0.2σ for a Gaussian probability distribution). The detection of these lags represents a significant advantage of the JAVELIN technique over traditional cross-correlation methods. We adopt the τ_multi as our final lag measurements, since the multi-line global fits provide well-constrained lags, properly treat covariances between the lags from different light curves, and utilize the maximum amount of information available in the data.

The analysis of NGC 4051 is especially difficult because the light curves exhibit low-amplitude variations. The lags in this object are also expected to be small, based on the AGN luminosity (Bentz et al. 2013) and a previous well-sampled RM experiment (Denney et al. 2009b). For Hβ, JAVELIN finds a definite lag near 2 days, consistent with the detrended ICCF approach. For Hγ, the ICCF method finds a lag consistent with zero, while the single-line JAVELIN fit finds a lag of 4.87 ± 0.18 days and the multi-line fit finds a lag of 2.40 ± 0.80 days (rest frame). The single-line fit results in a complicated multi-modal posterior distribution with smaller peaks at 15 and 25 days that are caused by aliasing. For example, the 25 day lag is probably caused by aligning the Hγ maximum near 6745 days with the local maximum in the continuum light curve at 6720 days (Figure 4). However, the multi-line fit shows a strong, dominant peak for Hγ at 2.40 days (rest frame). A probable explanation is that the Hβ light curve matches the overall shape of Hγ, but has stronger features against which to estimate a continuum lag—fitting both light curves simultaneously can therefore establish an Hγ lag with higher confidence. The problem with the Hγ light curve appears in a more serious form in the He ii λ4686 light curve, and JAVELIN finds a lag consistent with zero for this line.

Figure 10 shows the Hβ lags of our five objects as a function of luminosity, the so-called radius–luminosity (R–L) relation (Kaspi et al. 2000, 2005; Bentz et al. 2009, 2013). To estimate the luminosities, we first take the mean of the 5100 Å light curve and correct for Galactic extinction using the extinction map of Schlafly & Finkbeiner (2011) and a Cardelli et al. (1989) extinction law with R_V = 3.1. We then convert the flux to luminosity using the luminosity distances in Table 1. In the case of NGC 4051, which has a large peculiar velocity relative to the Hubble flow (z ∼ 0.002), we use a Tully–Fischer distance of 17.1 Mpc (Tully et al. 2008). This distance is uncertain by about 20%, and improving this measurement is an important step to investigate any discrepancies of this object from the R–L relation and to estimate its true Eddington ratio. For these purposes, an HST program has recently been approved to obtain a Cepheid distance to NGC 4051 (HST GO-14697; PI Peterson).

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The final values of $\lambda {L}_{5100\mathring{\rm{A}} }$ are reported in Table 1, along with the adopted Galactic values of $E(B-V)$. We find that our objects all lie close to, but slightly below (except for 3C 382), the R–L relation. The major systematic uncertainties are internal extinction in the AGN and host-galaxy contamination. Internal extinction may move the points farther from the R–L relation, but this effect is expected to be small. On the other hand, host-galaxy contamination can be very significant, especially for low-luminosity objects.

In order to correct for host contamination, we model high-resolution images of the targets and isolate the host-galaxy flux. This has previously been done for NGC 4051 (Bentz et al. 2006, 2013), and MCG+08-11-011, NGC 2617, and Mrk 374 were recently observed with HST for this purpose (HST GO-13816; PI Bentz). We also retrieved archival optical WFPC2 imaging of 3C 382 (HST GO-6967, PI Sparks), but the data are not ideal for image decompositions and we discuss the host-galaxy flux estimate for this object separately. A more detailed analysis of the HST GO-13816 data and image decompositions will be presented in future work (M. C. Bentz et al, in preparation). However, following the procedures described by Bentz et al. (2013), we made preliminary estimates of the host-galaxy contributions in the MDM aperture (150 × 50 aligned at position angle 0°). The results are given in Table 1 (uncertainties on these values are estimated at 10% and included in Figure 10). Applying this correction shows that host contamination accounts for the entire discrepancy between the observed luminosities and the R–L relation. The largest deviation from the R–L relation is Mrk 374, but the offset is only slightly greater than the 1σ scatter of the relation.

3C 382 resides in a giant elliptical galaxy and there may be a significant contribution from the host's starlight—several stellar absorption features are visible in the mean spectrum in Figure 5. In the archival HST images, the galaxy nucleus is saturated, hindering our ability to robustly remove the AGN flux and isolate the host's starlight. The main problem is that the Sérsic index of the host galaxy is degenerate with the saturated core and tends to drift toward unreasonably high values (n ≈ 7.6) when fitting the image in the same way as Bentz et al. (2013). Fixing the Sérsic index to more typical values (between 2 and 4) leads to host fluxes in the MDM aperture between 2.2 and 2.7 × 10⁻¹⁵ erg cm⁻² s⁻¹ Å⁻¹, about 77% of the observed luminosity ($\mathrm{log}\lambda {L}_{\mathrm{host}}=44.04$ to 44.12 [erg s⁻¹], after correcting for Galactic extinction). This estimate can be checked using the equivalent width (EW) of the prominent Mg absorption feature at 5200 Å rest frame (5460 Å observed frame). In our mean spectrum, we find an EW of 2.8 Å. In typical elliptical galaxy spectra, we find the EW is about 6.7–7.3 Å, depending on the continuum estimation and assumptions about the host-galaxy properties.⁵⁰ This implies that the featureless AGN continuum dilutes the absorption feature by a factor of 2.4–2.6, so that the host galaxy contributes approximately 40% of the observed luminosity. This rough estimate is a factor of two lower than the result from image decomposition, but the two values span the range of host-contributions from the other objects in our sample (42%–71% of the observed luminosity, see Table 1). We therefore adopt a host correction of (60 ± 20)% of the observed luminosity ($\mathrm{log}\lambda {L}_{5100\mathring{\rm{A}} }=43.98\pm 0.15$ [erg s⁻¹]), and we note that this estimate can easily be improved by obtaining unsaturated high-resolution images. The host correction moves 3C 382 away from the R–L relation, just beyond the 1σ dispersion. However, considering the large uncertainties, there does not appear to be any evidence that 3C 382 has an anomalous Hβ lag for its luminosity.

With the measurement of BLR velocity dispersions at a range of radii, it is possible to test if the BLR is virialized. Virialized dynamics predict V(r) ∝ r^−1/2, where the constant of proportionality depends on the SMBH mass and BLR inclination/kinematics. If the BLR is virialized, the virial products ${\sigma }_{L}^{2}c\tau /G$ derived from different line species should be consistent with each other, assuming similar geometries and dynamics for the line-emitting gas.

In Table 17, the maximum differences between $\mathrm{log}{\sigma }_{L}^{2}c\tau /G$ for each object are 3.3σ in MCG+08-11-011, 2.8σ in NGC 2617, 1.2σ in NGC 4051, and 0.4σ in Mrk 374. For NGC 4051 and Mrk 374, these differences are not significant. For MCG+08-11-011 and NGC 2617, the Hβ and He ii λ4686 virial products are marginally discrepant at about 2.5–3.3σ. We show these results Figure 11, which displays the linewidths σ_L as a function of lag τ_multi, and the relation ${\sigma }_{{\rm{L}}}\propto {\tau }_{\mathrm{multi}}^{-1/2}$ normalized by the value for Hβ. In this figure, we have applied a 0.13 dex uncertainty to both the lag τ and linewidth σ_L, representative of the characteristic systematic uncertainties. While the Hγ points generally agree with the Hβ relation, the He ii λ4686 points have very large offsets.

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There are many systematic issues that could account for these differences. As discussed in Section 4, the red wing of He ii λ4686 is blended with the blue wing of Hβ in both MCG+08-11-011 and NGC 2617. The He ii λ4686 velocity is therefore likely underestimated because we cannot follow its red wing underneath Hβ. The He ii λ4686 lags are also small compared to the monitoring cadence, and the lag is only marginally detected at 2.3σ in NGC 2617. Furthermore, the choice of line window and continuum interpolation can have a significant effect on the lag and linewidths. Finally, we must assume that the 5100 Å continuum light curve is a suitable proxy for the ionizing flux variations at extreme UV wavelengths. In NGC 5548, we found a ∼2-day lag between the far-UV and optical emission (Edelson et al. 2015; Fausnaugh et al. 2016). If a similar lag exists in these objects, it would change the He ii λ4686 virial products by a significant amount (0.3–0.4 dex), while the change in the Hβ virial products would be much smaller (0.05–0.11 dex). The effect of adding a 2-day UV–optical lag to the optical-line lags is shown in Figure 11, and the additional lag would reduce the discrepancies in the virial products to 1.3σ for MCG+08-11-011 and 2.0σ for NGC 2617. These AGNs have masses and luminosities similar to NGC 5548, so the existence of a UV–optical lag of this magnitude is very likely. Although a UV–optical lag affects the virial product and the characteristic size of the BLR, it does not affect the final mass estimate because the virial factor $\langle f\rangle$ is calibrated using the M_BH–σ_* relation (see Fausnaugh et al. 2016; Pei et al. 2017).

If the remaining discrepancies are real, they indicate different dynamics and geometries for the He ii λ4686 line-emitting gas compared to that of Hβ. This might be plausible, since He ii λ4686 is a high-ionization state line and may originate in very different physical conditions than the Balmer lines (for example, a disk wind). If He ii λ4686 has different dynamics than Hβ, it would be necessary to calibrate a different virial factor $\langle f\rangle$ for the He ii λ4686 line when calculating the SMBH masses. However, we cannot rule out systematic effects and it is unclear if the He ii λ4686 discrepancies are physical. If systematic issues do account for the discrepancies, then the dynamics of the BLRs in these AGNs would be consistent with virialized motion, as has been found for other AGNs (Peterson et al. 2004).

The Hβ light curves and line profiles have much higher S/N and very clear lags compared to both He ii λ4686 and Hγ, resulting in more reliable black hole masses. If we combine the virial products in Table 17 using statistical uncertainty-weighted average, the virial relation changes little, as shown in Figure 11 with the dashed lines. We therefore take the Hβ masses for our standard SMBH mass estimates.

Photoionization models make predictions about the structure of the BLR that can be tested with RM of multiple recombination lines. The locally optimally emitting cloud model (Baldwin et al. 1995) provides a natural explanation for the general similarity of AGN spectra and predicts radial stratification of the BLR—high-ionization state lines, such as He ii λ1640/4686 and C iv λ1549, should be primarily emitted at smaller radii than low-ionization state lines such as Hβ and Mg ii λ2798. Korista & Goad (2004, hereinafter KG04) used this model to predict that the responsivity of high-order Balmer lines should be greater than that of low-order lines (in the sense that ${\rm{H}}\gamma \gt {\rm{H}}\beta \gt {\rm{H}}\alpha$). KG04 also predicted that high-ionization state lines such as He ii λ4686 should have greater responsivity than all of the Balmer lines. Radial stratification of the BLR in NGC 5548 was first observed by Clavel et al. (1991), and has since been observed in several other objects (Peterson et al. 2004; Grier et al. 2013b). In addition, the expected trends of responsivity with ionization state/species have been confirmed in 16 AGNs by LAMP (Bentz et al. 2010; Barth et al. 2015).

We confirm these results for the four objects with multiple line light curves presented here. The He ii λ4686 lags in MCG+08-11-011 and NGC 2617 are less than 2 days, while the Hβ lags are 14.82 and 6.38 days, respectively, clearly indicating radial stratification. Furthermore, the fractional variability of the light curves, as measured by ${F}_{\mathrm{var}}$ (Table 4), is generally larger for Hγ than Hβ (or comparable for NGC 4051 and Mrk 374), while F_var for He ii λ4686 is always much greater than for the Balmer lines (although it is only slightly higher in NGC 4051). This implies that the relative line responsivities are He ii λ4686 ≫ Hγ > Hβ, in agreement with the photoionization models. We also find that the Hγ lags are slightly shorter than the Hβ lags within the same object (except for NGC 4051). KG04 showed that shorter lags are a natural consequence of the higher responsivity of Hγ compared to Hβ.

The formal definition of the responsivity of an emission line is

$$\begin{eqnarray}&&{\eta }_{\mathrm{line}}=\displaystyle \frac{{\rm{\Delta }}\mathrm{log}{F}_{\mathrm{line}}}{{\rm{\Delta }}\mathrm{log}{\rm{\Phi }}},\end{eqnarray} \tag{ 10 }$$

where F_line is the line flux and Φ is the photoionizing flux (KG04). The parameter η_line is therefore a measure of how efficiently the BLR converts a change in the photoionizing flux into a change in line emission. The ionizing flux Φ cannot be observed directly because these photons are at far-UV wavelengths (<912 Å). Therefore, we cannot measure ${\eta }_{\mathrm{line}}$ directly, but we can measure the relative responsivity ${\eta }_{\mathrm{line}1}/{\eta }_{\mathrm{line}2}={\rm{\Delta }}\mathrm{log}{F}_{\mathrm{line}1}/{\rm{\Delta }}\mathrm{log}{F}_{\mathrm{line}2}.$

We present rough measurements of the relative responsivity of Hβ, Hγ, and He ii λ4686 in Figure 12. For each object, we first removed the lags of each line from the corresponding light curve. We then matched observed points to the nearest day between the Hβ light curves and Hγ or He ii λ4686 light curves. The ratio ${\eta }_{\mathrm{line}}/{\eta }_{{\rm{H}}\beta }$ then corresponds to the slope of a linear least-squares fit to the data in the $\mathrm{log}{F}_{{\rm{H}}\beta }$–$\mathrm{log}{F}_{\mathrm{line}}$ plane.

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We find that ${\eta }_{{\rm{H}}\gamma }/{\eta }_{{\rm{H}}\beta }$ ranges from 0.74 to 1.44 and that ${\eta }_{\mathrm{He}{\rm{II}}}/{\eta }_{{\rm{H}}\beta }$ ranges between 0.73 and 6.23. NGC 4051, with ${\eta }_{\mathrm{He}{\rm{II}}}/{\eta }_{{\rm{H}}\beta }\sim 0.73$, is an outlier, probably caused by over-subtracting the continuum before integrating the line flux. For comparison, KG04 calculated η_line for a fiducial model of the BLR in NGC 5548, which includes an empirically motivated but ad hoc parameterization of the ionizing flux. From their Table 1, ${\eta }_{{\rm{H}}\gamma }/{\eta }_{{\rm{H}}\beta }$ ranges between 1.03 and 1.07, depending on the flux state of the AGN, while ${\eta }_{\mathrm{He}{\rm{II}}}/{\eta }_{{\rm{H}}\beta }$ ranges from 1.26 to 1.61. Thus, while our fits for ${\eta }_{{\rm{H}}\gamma }/{\eta }_{{\rm{H}}\beta }$ are in reasonable agreement with this fiducial model, the values of ${\eta }_{\mathrm{He}{\rm{II}}}/{\eta }_{{\rm{H}}\beta }$ are much larger than the model's prediction. The spread of ${\eta }_{\mathrm{line}}/{\eta }_{{\rm{H}}\beta }$ in our fits is also fairly large, which may indicate a diversity of photoionization conditions in the BLRs of different objects (perhaps due to harder or softer ionizing fluxes than assumed for NGC 5548).

Our estimates of the relative responsivities are sensitive to the total flux of the line light curves. For example, the sublinear slopes for ${\eta }_{{\rm{H}}\gamma }/{\eta }_{{\rm{H}}\beta }$ in NGC 4051 and Mrk 374 could be explained by missing variable line flux, perhaps in the wings of the line during low-flux states, or excess constant flux from the narrow emission lines or host-galaxy starlight. On the other hand, large values of ${\eta }_{\mathrm{He}{\rm{II}}}/{\eta }_{{\rm{H}}\beta }$ might be explained by contamination by Fe ii lines or misestimation of the continuum.