THE CASE FOR STANDARD IRRADIATED ACCRETION DISKS IN ACTIVE GALACTIC NUCLEI

We restrict the analysis in this section to objects in the Sergeev et al. (2005) sample, which gave high-confidence-level (P > 90%) results in the Chelouche & Zucker (2013) analysis of the V and R data. For those objects, solutions for τ_c, τ_l, and α were most secure, and τ_l values showed good agreement with independent constraints from spectroscopic reverberation mapping studies (Peterson et al. 2004; Grier et al. 2012, and references therein). Our subsample of objects therefore includes NGC 5548, NGC 4151, and NGC 3516 (Chelouche & Zucker 2013, see their Table 1 for the relevant data). De-trending of the light curves by a first degree polynomial was used throughout.

Solutions for τ_c, τ_l, and α are shown in Figure 2 for the various bands. The quoted values reflect on the mode of the distribution functions obtained from Monte Carlo simulations that introduce Gaussian measurement noise in accordance with the reported measurement uncertainties (Sergeev et al. 2005). Reported error bars reflect on ±68% percentiles around the mode value (Chelouche & Zucker 2013).

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Statistically significant τ_l values are detected in all three objects and in all bands but for the noisier B band (not shown). In particular, for a given object, the deduced values are consistent for the different bands, as may be expected if emission from similar locations in the BLR contributes to their flux. As noted in Chelouche & Zucker (2013, see their Table 1), these lags are also consistent with the spectroscopically measured Balmer time delays. This is indeed expected given that hydrogen emission dominates the non-power-law emission in those bands either through Balmer line emission or via Paschen lines and continuum emission (Figure 1).

As far as results for the B band are concerned, we note that higher-order Balmer line and iron blend emission are important contributors to the flux, and while the former originates from regions comparable to the Hα and Hβ emission regions (Bentz et al. 2010), the location of the latter is less secure with some studies placing it on scales comparable to or slightly larger than the Balmer line emission region (Bian et al. 2010; Hu et al. 2008); see also the new results of Barth et al. (2013) and Rafter et al. (2013), who find that the region is comparable in size to that which emits the Hα line (Bentz et al. 2010). For these reasons, in our analysis of the B-band data, we henceforth set τ_l to the spectroscopically measured Balmer line time delay (Chelouche & Zucker 2013, see their Table 1 and references therein), and determine τ_c and α for the B band under this prior (see Section 4.1.1).

The deduced values for α for the redder bands are of order 20% (NGC 5548 results in α ∼ 40%; see Figure 2). In comparison, the B-band data imply a smaller (5%–10%) relative contribution of a lagging emission component to the band. Indeed, a qualitative spectral decomposition of the Glikman et al. (2006) composite spectrum suggests that the contribution of non-power-law emission to the bands is of the same order¹ (Figure 1).

Our τ_c measurements relative to the V band show a qualitative trend whereby τ_c increases with wavelength from negative values for the B band to positive lags for the redder bands. This is most notable for NGC 4151 and less so for NGC 3516 due to the larger uncertainties. Results for NGC 5548, while formally negative, are consistent with a zero lag in all bands, with only the upper uncertainty envelope hinting on a possible increase of τ_c with wavelength.

4.1.1. Setting Priors on τ_l

The consistency between the τ_l values deduced for the different bands, the agreement between those and the spectroscopic time-lag measurements for the Balmer emitting region (Figure 2 and Chelouche & Zucker 2013), and the reasonable results for τ_c and α obtained for the B band when using priors motivate us to consider a refined analysis approach, where the number of degrees of freedom is reduced. In particular, we henceforth set τ_l to the spectroscopically measured Balmer emission line lag (Chelouche & Zucker 2013, see their Table 1 for the adopted values, and note that we neglect the measurement uncertainty on τ_l, as well as the weak inverse trend between the time delay and the increasing line order; Bentz et al. 2010), and constrain the values for τ_c and α under this prior in the remaining two-dimensional phase space for all bands. The results are shown in Figures 3 and 4 for the three objects in our subsample.

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Generally, the results obtained by setting priors on τ_l are consistent with those deduced without priors, yet the uncertainties are reduced, and higher significance τ_c measurements are obtained. Specifically, the deduced τ_c values are in qualitative agreement with irradiated accretion disk model predictions, where the lag increases with wavelength as one views larger emitting regions. By and large, the deduced τ_c values are consistently below those identified by inter-band time delays (Cackett et al. 2007; Sergeev et al. 2005, most notably for NGC 5548; see our Figure 3), which are contaminated by emission lines' signal (Chelouche & Zucker 2013), and are in better agreement with those obtained for a similar luminosity object using spectroscopic means (Collier et al. 1998, see also Section 4.2).

The solutions for α are now better defined and show fair agreement with qualitative spectral decomposition estimates (Figure 4 and more in Section 4.2). For example, NGC 3516 shows a maximal α value in the R band, which can be attributed to the dominant contribution of Hα (Figure 1). Results for NGC 4151 imply an overall reduced contribution of emission lines to the flux (more accurately, flux variance) compared to NGC 3516, but that which monotonically rises toward the redder parts of the spectrum. There are various possible reasons for that, among which a BLR that varies less on the relevant timescales, and/or continuum emission that varies less in the redder parts of the spectrum, or is suppressed altogether due to disk truncation (it is not possible to robustly test either of these scenarios using the present data set). Lastly, we note the case of NGC 5548 that exhibits α values that are in excess of what is expected from spectral decomposition, and in addition leads for formally negative (yet consistent with zero) lags for the redder bands. Together, these facts lead us to doubt the validity of the τ_c and α solution in this case, and we exclude this object from further analysis.

We next analyze the remaining nine objects in the Sergeev et al. (2005) sample using spectroscopic priors on τ_l (Table 1 in Chelouche & Zucker 2013). Results for most objects are consistent with τ_c rising monotonically with wavelength from negative values for the B band to positive values for the redder bands (Figure 5). In particular, results for NGC 7469 are in good agreement with the spectroscopic lag measurements of Collier et al. (1998). NGC 3227 and Akn 120 are exceptions as τ_c for the B band is marginally positive, in contrast to naive model expectations where the bluer bands should lead the redder ones. The reason for that is currently unclear, and may have to do with the particular light-curve behavior observed, sampling, or residual signal from emission lines. Results for the R1 and I bands are largely consistent as the centroid wavelengths of both bands are quite similar² (Figure 1).

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Considering the statistical properties of the entire sample (Figure 6), we note that the median τ_c is consistent with being monotonically rising with wavelength, i.e., with geometrically thin accretion disk model predictions where τ_c(λ) ∼ λ^4/3 (Collier et al. 1998).³ In comparison, the best-fit curve obtained using the median results of Cackett et al. (2007) is much steeper, with an effective linear slope that is ∼2 times greater than the one found here. As discussed above, this is likely because of BLR emission contamination of the signal in cross-correlation analyses (Chelouche & Zucker 2013).

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Turning our attention to α, we note that it is, typically, smallest for the B band, peaks in the R band, and somewhat declines, yet remains finite also for the R1 and I bands (Figure 5). This behavior is expected from a naive spectral decomposition of the Glikman et al. (2006) composite into power-law and non-power-law emission components (Figure 4). Particular objects (e.g., NGC 3227) show, however, a more monotonic increase of α with wavelength, as discussed in Section 4.1.1. Further investigation into the different α behavior observed in different objects is outside the scope of this work.

We first consider the relation between the lag and the object luminosity, both of which are independent observables in our study. Figure 7 shows the best-fit relation to the data, $\tau _c^0\propto L_{\rm opt}^{0.53\pm 0.10}$ (error bars were obtained by means of Monte Carlo simulations that take into account the errors on individual $\tau _c^0$ measurements, but do not consider the uncertainty on the luminosity). In the standard theory of accretion disks, as applies to AGNs (Shakura & Sunyaev 1973; Davis & Laor 2011), the optical spectrum is self-similar (L_ν ∼ ν^1/3, where ν is the frequency), and hence results in $L_{\rm opt} \sim (M_{\rm BH} \dot{\mathcal {M}})^{2/3}$, where $\dot{\mathcal {M}}$ is the effective mass accretion rate (allowing for irradiation; see the Appendix). The light crossing time of the optical emitting region in such a disk is $\tau _c^0\sim (M_{\rm BH} \dot{\mathcal {M}})^{1/3}$ (Collier et al. 1998). This implies that for a thin irradiated self-similar accretion disk one expects $\tau _c^0\sim L_{\rm opt}^{1/2}$, which agrees well with the observed relation.

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One may wonder whether our $\tau _c^0$ measurements on subsampling times, τ_s, are at all meaningful (e.g., NGC 4051 has a formal lag of ∼0.2 ± 0.1 while the minimum visit interval is ∼0.8 days). The answer depends on (1) the number of data points and (2) the power density spectrum of the quasar. In particular, red power density spectra, which have little power at high frequencies (e.g., ${>}1/\tau _c^0\,{>}\, 1/\tau _s$), may result in robust continuum lag measurements provided enough data with good S/N exist. In fact, this justifies the use of linear interpolation schemes for evaluating correlation functions in the general context of AGN reverberation mapping (Cackett et al. 2007; Kaspi et al. 2000; Sergeev et al. 2005). Nevertheless, sampling is known to affect time-lag measurements (Grier et al. 2008), and it will be of importance to repeat this experiment with better-sampled data, especially for the lower luminosity objects, considering the typical values for $\tau _c^0$ determined here.

We note that while the minimum cadence of the observations is similar for all objects in the Sergeev et al. (2005) sample, the median sampling, which may be a more meaningful quantity as far as time-lag measurements are concerned, does show a weak trend with luminosity (Figure 7). Nevertheless, were our time-lag measurements merely reflecting on the slightly different sampling used, $\tau _c^0$ would have shown a mere factor of <2 in range, in contrast to our findings. With that being said, it will be of importance to repeat this analysis for a sample of objects having a range of luminosities with luminosity-independent sampling, to verify that residual biases are indeed eliminated.

We note that Mrk 509 is consistently an outlier in our sample; whether this is due to extreme inclination (see below) or, more likely in our opinion, due to an erroneous lag measurement (note the poor fit in Figure 5) is yet to be determined.

The fact that $\tau _c^0 \propto L_{\rm opt}^{1/2}$ does not necessarily imply that viscous heating (perhaps with some irradiation) is primarily responsible for the bulk of the optical emission. For example, the light intensity of a varying point source that scatters off a uniform extended surface could result in a similar size–luminosity relation. To show that the irradiated thin accretion disk physics indeed provides a viable interpretation of the data, we will next relate the measured time delays to the optical luminosity of the source within the framework of a Shakura & Sunyaev (1973) accretion disk model.

Note that for low-amplitude radiative perturbations of a disk carrying an effective mass accretion rate, $\dot{\mathcal {M}}$ (Collier et al. 1998, and the Appendix),

$$\begin{equation} \tau _c^0\simeq 0.7 \left(\frac{M_{\rm BH}\dot{\mathcal {M}}}{{10^6{\,M}_\odot ^2 \,{\rm yr}^{-1}}} \right)^{1/3}\,{\rm days}, \end{equation} \tag{ 2 }$$

where the exact normalization depends on which property of the continuum transfer function is being traced (for reverberation mapping algorithms that effectively measure the centroid of the transfer function, the above normalization holds irrespectively of the disk inclination⁷). Our normalization of Equation (2) is smaller by a factor ≃ 2 than the one used by Collier et al. (1998, allowing for the different definitions used in each work), and is supported by numerical calculations outlined in the Appendix. It is also in better agreement with the revised values adopted in Collier (2001).

The optical flux in some AGNs has been known to vary significantly over long, ≫1 day, timescales (Uttley et al. 2003), which are, nevertheless, shorter than dynamical timescales associated with the optically emitting region. If due to variations in the irradiation level, then the disk would effectively "breath" during the time series, increasing/decreasing its size in high/low flux states, as $\dot{\mathcal {M}}$ varies (Equation (2) and the Appendix). Therefore, as one extracts the disk size from the full light curves, one actually measures some average value over the time series (cf. the corresponding effect for the BLR; Peterson et al. 2002). This average is, however, difficult to define, and hence to accurately interpret, since it depends on how the signal is accumulated by the correlation function and therefore on the light-curve properties. Specifically, the particular variability pattern of the AGN, the sampling used, and the S/N of the data are all relevant factors in determining the exact value of the average lag. For the Sergeev et al. (2005) sample, we find that flux variations could affect the effective disk size at the ∼10% level for most objects (with the most pronounced effect being at the ∼17% level for 3C 390.3). Such uncertainties should be borne in mind when quantitatively interpreting the data, and better data sets are required to shed light on their potential importance.

Inverting Equation (2), one can use the measured lags to determine the product $M_{\rm BH} \dot{\mathcal {M}}$, which, in turn, sets the optical luminosity for AGNs (Bian & Zhao 2003; Collin et al. 2002; Laor & Davis 2011, and the Appendix). This leads to the total optical disk luminosity being

$$\begin{equation} L_{\rm opt}^{\rm model} \simeq 3\times 10^{43} \left(\frac{\tau _c^0}{{\rm days}} \right)^2\,{\rm erg\,s^{-1}}. \end{equation} \tag{ 3 }$$

To be able to compare model predictions to observations, whose reported values assume isotropic emission of the source, we note that $L_{\rm opt}^{{\rm model,iso}}=2L_{\rm opt}^{\rm model}\,{\rm cos}(i)$, where i is the inclination angle. This means that for sources observed with inclination angles i > 60°, the implied isotropic luminosity will be lower than the true luminosity of the source. Nevertheless, for type I sources, whose continuum emission is not obscured by the putative torus, it is believed that, generally, i  <  60°, which is also consistent with the findings of microlensing studies (Poindexter & Kochanek 2010).

Figure 8 shows model predictions for the isotropic optical luminosity (i = 0° is assumed) versus the measured value. It is important to re-emphasize that, a priori, it is not clear that Equation (3) should predict a luminosity similar to the observed one as our time-lag measurements are independent of the object luminosity (so long as the AGN is not too faint to be observed). The fact that a 1: 1 relation between the measured and predicted quantities is roughly obtained (note the different 1: 1 relations plotted for various object inclinations in Figure 8) provides strong evidence for the prevalence of standard thin irradiated accretion disks in AGNs, at least as far as their optical emission properties are concerned. Furthermore, it implies that our time-lag measurements are probably in the right ballpark, and that type I AGNs are not highly inclined sources, as expected by theory. For example, had our time-lag measurements been a factor ∼2 larger than observed (cf. Cackett et al. 2007 and our Figure 6), the predicted luminosities would have been a factor ∼4 larger, implying either that AGNs in the Sergeev et al. (2005) sample are highly inclined, or that accretion disks are intrinsically inefficient emitters, in contrast with the underlying physical assumption of geometrically thin disks (Shakura & Sunyaev 1973). With that being said, one should be careful to not overinterpret the data, especially at the low-luminosity end, where undersampling could lead to overestimated lags and hence a bias (see above).

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Our results are consistent with the notion that the bulk of the optical luminosity in AGNs is emitted by standard, geometrically thin, irradiated accretion disks. In contrast, using data for lensed quasars, Morgan et al. (2010) concluded that accretion flows are inconsistent with the bulk of the (rest-UV) emission originating in standard disks, as they are subluminous given their microlensing-implied sizes (assuming they radiate as blackbodies with T ∼ r^−3/4). In particular, they found that the size of their rest-UV emitting regions is, on average, a factor ∼4 larger than predicted by theory for plausible values of the radiative efficiency, object inclination, and BH mass. Crudely, had the same effects been present here, our predicted luminosities would have been ∼16 times greater than observed (Equation (3)), which is, generally, not the case.⁸ Whether this indicates a departure from self-similarity in the inner disk regions of AGNs, a genuine difference between quasars and AGNs, or systematic effects (and/or inadequate interpretation of the signal) in either of the methods requires further investigation. In what follows, we shall assume that geometrically thin accretion disks provide a viable physical explanation for the bulk of the optical emission in AGNs, and consider the physical implications of our results.

Our time-lag measurements, as interpreted by standard thin (infinite) accretion disk models, may be used to measure the effective mass accretion rate, $\dot{\mathcal {M}}$, independently of the luminosity, in objects for which M_BH is known (Equation (2)).⁹ Typical $\dot{\mathcal {M}}$ values for objects in the Sergeev et al. (2005) sample are shown in Figure 9, where the results hover around 0.01–0.1 M_☉ yr⁻¹. When contrasted with the accretion rates of the Palomar–Green (PG) objects (Davis & Laor 2011), Seyfert 1 galaxies tend to have $\dot{\mathcal {M}}$ values that are smaller by about 1–2 orders of magnitude, which is consistent with their luminosities being similarly lower than bona fide quasars. Comparing our results for the mass accretion rate to those of other Seyfert 1 galaxies from Raimundo et al. (2012), based on optical luminosity considerations (Bechtold et al. 1987), we find qualitative agreement with their findings. We note, however, that the true mass accretion rate is somewhat lower than deduced here, as $\dot{\mathcal {M}}$ includes the effects of irradiation; this is also true for the Davis & Laor (2011, see also Raimundo et al. 2012) results (see the Appendix).

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An interesting property of our results is that the deduced $\dot{\mathcal {M}}$ values (apart from three objects, including Mrk 509) span only about a factor of ∼5 in range, and could even be similar given the measurement uncertainties (mainly on M_BH). In contrast, the AGNs in our sample span almost two orders of magnitude in luminosity. Whether this implies a common triggering mechanism for low-luminosity type I AGNs, which sets a rather uniform mass accretion rate across a broad range of BH masses, or whether this result vanishes as better data and more statistics are accumulated remains to be determined. It is, however, interesting to note that a similar non-dependence of the (median) mass accretion rate on M_BH has been previously reported by Davis & Laor (2011, see our Figure 9), for the PG quasars, over a similar range of BH masses.

It is furthermore possible to estimate the radiative efficiency of Seyfert galaxies, $\eta \equiv L_{\rm bol}/\dot{\mathcal {M}}c^2$, provided the bolometric luminosity of the source, L_bol, may be reasonably estimated from the limited spectral range probed. To this end, we first assume that L_bol/L_opt ≃ 9, and is independent of M_BH, as was assumed for the PG sample of quasars (Davis & Laor 2011). Alternatively, we assume that L_bol/L_opt ∼ 9(M_BH/10⁸ M_☉)^−0.5, which is consistent with the findings of Raimundo et al. (2012, see also Scott et al. 2004) for their sample of AGNs. We plot η(M_BH) in Figure 9 using the above bolometric corrections, and conclude that the radiative efficiency is indeed consistent with the typical efficiencies from geometrically thin accretion disk models (Narayan & Quataert 2005). The results for a constant bolometric correction trace well the findings of Davis & Laor (2011) for the PG sample of quasars. This means that, under the same assumptions, the radiative efficiency, η, for PG quasars and for Seyfert 1 galaxies behaves in a similar manner with the BH mass, and is independent of the mass accretion rate.¹⁰ Using a mass-dependent bolometric correction does not appreciably alter the above conclusions.

Our findings provide tentative evidence for η being a rising function of BH mass, as previously reported by Davis & Laor (2011) based on SED arguments. With that being said, better bolometric corrections for individual objects, as well as better time-delay measurements for the least massive AGNs, are required to firmly establish such a trend, and, if real, its origin.

In this study we were able to measure, for the first time in a non-biased way and with a high success rate, continuum-to-continuum time delays in 11/12 low-luminosity AGNs. To securely determine the time delays associated with an irradiated accretion disk, high-cadence, good-S/N observations are required, and prior knowledge of the BLR size could be advantageous in narrowing down the parameter space, thereby leading to higher-confidence results.

Evidently, minimal benchmarks for a successful reverberation campaign are provided by the Sergeev et al. (2005) sample. Nevertheless, given the shorter delays deduced here compared to the Sergeev et al. (2005) findings, higher-cadence observations, especially for low-luminosity objects, are preferred. With upcoming photometric surveys, such as the Large Synoptic Survey Telescope (especially their deep-drilling fields), it may be possible to obtain much higher quality data for many more (bright) objects, potentially transforming the field.

In the long term, it may be possible to look for deviations from simple infinite self-similar disk models, which could shed light on the role of outflows in those systems, as well as on the accretion rate in the inner disk regions, which ultimately feeds the BH. Further, it may be possible to probe the far outskirts of the disk, where it becomes self-gravitating, and may join with other components of the AGN, such as the BLR. In addition, it may be possible to determine the inclinations of accretion disks by comparing their apparent luminosities to their predicted ones. Lastly, reverberation mapping of the accretion flows in numerous AGNs could reveal the different modes of accretion prevailing in those systems (e.g., slim versus thin disk accretion) with implications for BH growth rates.

Once accretion disk models are quantitatively tested and verified and reddening corrections and the host contribution to the flux adequately estimated, it may be possible to use quasars as standard candles, as envisioned by Collier (2001) and Cackett et al. (2007, and references therein). Interestingly, as our deduced lags are a factor ∼2 smaller than those considered by Cackett et al. (2007), with all other quantities held fixed at the values adopted by these authors, our deduced value for H₀ based on their formalism (see their Equation (15)) would be in the range 42–88 km s⁻¹ Mpc⁻¹ and in better agreement with concordance cosmology. The full treatment of this topic is beyond the scope of this paper.