EVIDENCE FOR AN INTERMEDIATE LINE REGION IN ACTIVE GALACTIC NUCLEI's INNER TORUS REGION AND ITS EVOLUTION FROM NARROW TO BROAD LINE SEYFERT I GALAXIES

First, we focus on Hβ lines; several examples are shown in Figures 1 (panels a1, b1, and c1) and 4(b). The statistical analysis is presented in Figure 3 (panels a1, b1, and c1). The VBGC has a much larger FWHM than the whole line when FWHM of the whole line is small. The FWHM ratios range from 1.5 to 4 in the NLS1 group which lie on the left of Figure 3 (panel a1). The intensity ratio of the VBGC to the whole line is about 0.6 in these objects. This means that for NLS1s the IMGC has an intensity comparable to the VBGC, and consequently the FWHM of the whole line of NLS1s is dominated by the IMGC. With FWHM increasing, the IMGC becomes weaker and finally disappeared, which can be seen in Figure 3(b1), in which the intensity ratio of the VBGC to the whole line reaches unity when the FWHM of the whole line reaches about 5000 km s⁻¹. Naturally, the FWHM ratio of the VBGC to the whole line also reaches unity. The whole line can thus be simply described by a single VBGC and the VBGC becomes totally dominant. This trend agrees with that obtained by Hu et al. (2008), who have shown that most of the AGNs with very large black hole masses (normally with largest FWHM) do not need two components for their broad Hβ lines. Our result also agrees with Collin et al. (2006), who found that NLS1s seem to have a prominent narrower component (the IMGC identified in our work) on top of a broad Gaussian profile (the VBGC identified in our work).

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In Figure 3 (panel c1), the FWHM ratio of the IMGC to the VBGC becomes larger with increasing FWHM. It ranges from 0.2 to 0.6, which is another reason for the FWHM of the whole line reaching the FWHM of the VBGC. It indicates that the IMGC also becomes broader when it becomes weaker. The points that lie on y = 0 of the bottom picture represent the objects which have only a single Gaussian component. The three points with maximum FWHM (but still with two Gaussian components) fall off the trend slightly, due to perhaps the large uncertainties in fitting when the intensity of the IMGC becomes very low.

Compared to Figure 2, the correlations in Figure 3 are much tighter, suggesting that the two-component decomposition is physically meaningful and not just a mathematical convenience. Moreover, the systematical evolution shown in Figure 3 suggests that the two components are physically distinct, and both components respond to a common source.

Figures 1 (panels a2, b2, and c2) and 4(a) are several decomposition examples for Hα lines. They behave similarly when the FWHM is small. However, when the FWHM is large, the IMGC is very broad and strong. Statistical analysis is shown in Figure 3 (a2, b2, and c2) in a similar manner to Hβ lines. The obvious difference is that the number of points near unity in panel (b2) is less than that in panel (b1) for Hβ lines; only a weakly increasing trend can be seen in panel (b2). This means that the IMGC is still strong even when the FWHM reaches about 5000 km s⁻¹.

Hγ lines behave similarly to Hβ lines as shown in Figures 1 (panels a3, b3, and c3) and 4(c); when the FWHM is very large, only a single Gaussian component (the VBGC) is required to fit the profile. However, when the FWHM is very small, a large number of Hγ lines loose the VBGC, i.e., only the IMGC is required to fit the line, because most of their FWHM are very close to the IMGCs of their Hβ lines. The points that lie on the x-axis in Figure 3 (panel b3) are the cases when the Hγ lines are described by a single IMGC; an example is presented in Figure 4(c). However, some of the single Gaussian component Hγ lines have FWHM neither close to the VBGC nor close to the IMGC of the corresponding Hβ lines, therefore we exclude these points in our analysis. The loss of the VBGCs may be caused by the weak intensity of Hγ, since its VBGC is particularly weak when the FWHM is small, which can cause confusion in the continuum subtraction. The gap pointed by an arrow in Figure 3 (panel b3) also reveals a sudden decrease of the intensity of the broad Gaussian component when it is very weak.

Previous works suggested that a highly significant correlation exists between FWHM(Hβ) and source luminosity (e.g., Joly et al. 1985; Corbett et al. 2003). More recent work with a larger sample that spans an FWHM range from 1000 to 16,000 km s⁻¹ showed that the correlation is not so strong but still statistically significant; the correlation in this large sample is likely driven by the minimum FWHM trend (Marziani et al. 2009). Figure 5(a) presents this correlation in our sample; the correlation is more obvious in this smaller sample than in Marziani et al. (2009), and it is very similar to the correlation in Joly et al. (1985). The extremely broad Hβ (FWHM > about 7000 km s⁻¹) may have a rather different line profile and may originate from a different physical region from the less broad Hβ (e.g., Strateva et al. 2003; Eracleous & Halpern 2003), therefore the analysis with only mediate broad (FWHM < about 7000 km s⁻¹) Hβ with the same line profile seems to be more meaningful. Figure 5(b) shows that the logarithm of the FWHM of the VBGC or the IMGC increases with the logarithm of luminosity in a more linear way than the FWHM of the whole line. This may indicate that the FWHM of the VBGC or the IMGC is more physical than the FWHM of the whole line. On the other hand, Figure 5(b) also shows that the FWHM of the IMGC increases with luminosity much faster than the VBGC; the two components have a trend to merge into one. This trend will be confirmed by another way in Section 4.1. Therefore, FWHM evolution also supports the picture that the BLR is composed of two physically distinct regions.

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We assume that the two Gaussian components come from two distinct regions, and both of the these two regions (VBLR and IMLR) are bounded by the central black hole's gravity. We therefore have

$$\begin{equation} f_1^2\frac{V_{\rm { VBLR}}^{2}}{R_{\rm { VBLR}}}=\frac{GM_{\rm { BH}}}{R_{\rm { VBLR}}^{2}}, \end{equation} \tag{ 1 }$$

and

$$\begin{equation} f_2^2\frac{V_{\rm { IMLR}}^{2}}{R_{\rm { IMLR}}}=\frac{GM_{\rm { BH}}}{R_{\rm { IMLR}}^{2}}. \end{equation} \tag{ 2 }$$

Equations (1) and (2) lead to

$$\begin{equation} R_{\rm { IMLR}}=\left(\frac{f_1V_{\rm { VBLR}}}{f_2V_{\rm { IMLR}}}\right)^{2}R_{\rm { VBLR}}=C_0\left(\frac{V_{\rm { VBLR}}}{V_{\rm { IMLR}}}\right)^{2}R_{\rm { VBLR}},\\ \end{equation} \tag{ 3 }$$

where

$$\begin{equation} C_0=\left(\frac{f_1}{f_2}\right)^2.\\ \end{equation} \tag{ 4 }$$

We assume that C₀ is a constant for all sources, because currently we cannot determine the exact geometry and velocity distribution of gases in the IMLR. The validity of this assumption and the exact value of C₀ needs to be determined with further studies of more observation data. For convenience we simply redefine R_IMLR = R_IMLR/C₀ for the rest of this paper, i.e., we only consider the special case when C₀ = 1 (see Section 5.2 for a tentative supporting for this reason rather arbitrary choice).

RM is used to measure the radius of the BLR, which is related to an AGN's continuum luminosity by the following empirical radius–luminosity (R–L) relation (Kaspi et al. 2005):

$$\begin{equation} \log {\frac{R_{\rm { BLR}}}{\rm {lt-days}}}=(0.69 \pm 0.05) \log {\frac{\lambda L_{\lambda }(5100 {\AA})}{\rm {erg}\; \rm {s}^{-1}}}-29.0\pm 2.2, \end{equation} \tag{ 5 }$$

or the starlight-corrected R–L relation (Bentz et al. 2006):

$$\begin{equation} \log {\frac{R_{\rm { BLR}}}{\rm {lt-days}}}=(0.518 \pm 0.039) \log {\frac{\lambda L_{\lambda }(5100 {\AA})}{\rm {erg}\; \rm {s}^{-1}}}-21.2\pm 1.7. \end{equation} \tag{ 6 }$$

We use Equation (6) to calculate R_BLR throughout this work, unless indicated otherwise. Equation (5) is only used in Section 4.4 where a comparison is made.

Because RM is used to measure the radius of the BLR, one might expect that it would normally measure the radius of the innermost emission-line region, i.e., the VBLR in our model (see the discussion on this issue in Section 5.2). Therefore, the radius calculated from R_BLR ∼ L5100 relation is taken as R_VBLR. Consequently, we can calculate the radius of the IMLR from Equation (3). Both Hα and Hβ lines can be used to do this calculation. Because they seem to behave slightly differently in the line evolution, R_IMLR is obtained from Hα and Hβ lines separately. Figure 6 is the evolution of R_VBLR and R_IMLR with FWHM. IMLR radii derived from Hα and Hβ lines have only subtle differences. The IMLR radius just varies around a constant with increasing FWHM, i.e., does not have a systematic increase or decrease. On the other hand, VBLR radius becomes larger with increasing FWHM. VBLR and IMLR are clearly separated with FWHM < 2500 km s⁻¹ as we can see in Figure 6. The circles filled with dots represent the objects whose lines are fitted well with a single (very) broad Gaussian component; they appear in Figure 6 with the largest FWHM.

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The evolution of R_VBLR and R_IMLR with luminosity and black hole mass are shown in Figure 7. The radius of the IMLR increases more slowly than VBLR with black hole mass and luminosity. The two emission regions have a trend to merge into one with higher luminosity or larger black hole mass; it is consistent with the emission lines' behavior as shown in Section 3.4. The points that represent objects with zero IMGCs also mainly lie on the large mass and large luminosity side. The slopes of these relations will be discussed in Section 4.4. Although most of the Hα lines still need two Gaussian components to fit, there are three of them which only need one Gaussian component in our sample. The two Gaussian components in Hα lines also show a trend to merge into one single Gaussian component with the FWHM becoming larger, though the trend is not as obvious as is for Hβ lines. Once again the systematically different trends of evolution of R_VBLR and R_IMLR are consistent with our assumption that they are physically two distinct emission regions.

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It is possible that density, temperature, and geometric shape of the two regions (VBLR and IMLR) are different, then the collisional effect and different ionization energies of the three Balmer lines may cause different Balmer decrement in the two regions. Here, we show the Balmer decrement of the whole line, the VBGC and IMGC in Figure 8. There are two sources whose Hα or Hβ or both do not fit well with the double Gaussian model; these two points are excluded in Figure 8(a) when calculating the intensity ratio of Hα to Hβ. Similarly, there are 16 sources whose Hβ or Hγ or both do not fit well; these points are excluded when calculating the intensity ratio of Hβ to Hγ. The points excluded in Figure 8(a) are also excluded in Figures 8(b) and (c). In addition, 15 sources' Hγ do not have broad Gaussian; these points do not appear in the intensity ratio of Hβ to Hγ in Figure 8(b). Fourteen sources' Hα or Hβ or both only need a VBGC to fit (i.e., they do not have the IMGC), so these sources are not included in Figure 8(c) when calculating the Hα-to-Hβ ratio. Similarly 22 sources' Hβ or Hγ or both only have VBGC, and are thus excluded from Figure 8(c) when calculating the Hβ-to-Hγ ratio. In summary, there are 88, 88, and 76 points used for calculating Hα-to-Hβ ratio in Figures 8(a), (b), and (c), respectively; correspondingly, there are 74, 59, and 52 points in Hγ-to-Hβ ratio.

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In the VBLR, Hα/Hβ is 2.54 and Hγ/Hβ is 0.37, which are consistent with a pure H model with T = 15, 000 K, Ne = 10¹⁰ cm⁻³ (Krolik et al. 1978). In the IMLR, Hα/Hβ is 4.78, which is much higher than that in the VBLR, consistent with previous results (Netzer & Laor 1993; Mullaney & Ward 2008). These can be caused by the collisional effect which can lead to an effective emission increase for Hα but not for Hβ or Hγ (e.g., Osterbrock 1989). This result suggests a higher gas density in the IMLR. It has also been argued that dust in the emission region can cause higher Balmer decrement (Binette et al. 1993). Therefore, it also suggests that the IMLR may be contaminated by dust. On the other hand, the Hγ-to-Hβ ratio is also slightly higher in the IMLR (0.41) than that in the VBLR (0.37). As Hβ and Hγ are produced from similar process, the slightly higher Hγ-to-Hβ ratio is not well understood.

In Figure 9, we show the correlations between the first three Balmer lines for the IMGC, i.e., between the Hα and Hβ lines, and between the Hγ and Hβ lines. The linear correlations between them indicate that the IMGCs for all these three lines originate from physically connected regions, even if not exactly from the same region. In Figure 10, we show the distributions of FWHM differences between the three lines for the IMGC. The FWHM of Hα and Hγ is offset systematically by around −200 km s⁻¹ and +200 km s⁻¹ around that of Hβ, respectively. This suggests a stratified geometry for the IMLR, where Hγ, Hβ, and Hα lines are produced at increasing radii respectively. The FWHM of the whole line, Hα is also systematically larger than Hβ (Greene & Ho 2005; Shen et al. 2008). It is consistent with the RM result (Kaspi et al. 2000) which showed that the radius of the BLR obtained from Hα is larger than that of Hβ in most of the sources. Although there is no difference in the ionization degree of Hα, Hβ, and Hγ, the above result can be explained by the fact that the inner skin of the torus is the IMLR, therefore, it has more dust in the region with larger radius. There are mainly two processes. First, dust reddening will make the Hγ produced in the region much closer to the torus to be extincted most significantly, the Hβ will suffer less extinction and Hα will be least extincted. Therefore, the average radius of the Hα region will be larger than Hβ and Hγ. Second, collisional excitation contributes a lot to Hα line but not significantly to Hβ and Hγ line, so the region much closer to the torus with a higher density will produce relatively more Hα. This also causes the FWHM of Hα to be smaller than Hβ and Hγ.

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The relation between the irradiation region and the central continuum is important, since the emission lines are thought to be mostly caused by photoionization. Figure 11 is plotted based on the analysis of Hβ lines. The Baldwin effect can be seen in the bottom plot. The Pearson correlation coefficient (PCC) is −0.39. We bootstrap it 100,000 times and obtain 10 PCC larger than 0, so PCC is smaller than 0 at 99.99% confidence level. The Pearson rank correlation coefficient (PRCC) is −0.48, at 99.9% confidence level smaller than 0 using the same method as the confidence level calculation of PCC. However, no Baldwin effect is seen in the top (Pearson correlation coefficient is −0.1) or in the middle plot (Pearson correlation coefficient is 0.1). It means that the equivalent width (EW) of the IMGC becomes smaller when the AGN becomes brighter, whereas the EW of the VBGC does not change with the continuum luminosity. This phenomenon has also been seen by Marziani et al. (2009). Since EW reflects the covering factor of the emission region, the above result supports the scenario that the VBLR is nearly spherical but the IMLR has a flattened geometry. Therefore, in the absence of any other knowledge of other parameters of the gas, we can use the higher EW as supportive (albeit not conclusive) evidence for a higher covering factor. There is also evidence that the dust torus has a smaller covering factor with higher luminosity (Wang et al. 2005). We therefore suggest that the cause of the slight Baldwin effect of the IMLR can be the same, i.e., the innermost region of the dusty torus can be sublimated away by the strong irradiation of the AGN. Therefore, we can infer that the IMLR and VBLR have different geometries.

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Based on the above analysis, a picture of the IMLR has emerged. In comparison with the VBLR (the traditionally called "BLR" with a near spherical structure in virialization equilibrium), it has a larger radius and higher density, contains more dust, is more flattened, and is thus consistent as being the inner boundary region of the dust torus of AGN. As shown in Figure 7, the radii of both VBLR and IMLR are primarily determined by the continuum luminosity of the AGN. R_BLR obtained with Equation (6) is used as the radius of the VBLR there. Then R_IMLR ∝ L^0.37±0.06_opt is derived from Equation (3) as we have shown in Figure 7(a). The stratified geometry of the IMLR, as inferred from Figure 10 in Section 4.2, is also consistent with the above picture.

The receding velocity of torus with luminosity can be also obtained from the analysis of type I AGN fraction. A relation between the covering factor C and L5100 can be obtained from Maiolino et al. (2007): $\log {C}=-0.18\log {\left(\frac{L5100}{\rm erg\;s^{-1}}\right)}+7.4$, consistent with Wang et al. (2005). If R_IMLR is used as the radius of the inner torus, then $C=4\pi \sin {\theta }=4\pi \frac{h}{h^2+R^2_{\rm IMLR}}$, and the height of the inner torus h can be calculated from this relation, as shown in Figure 12. Clearly the relationship between h and R_IMLR can be fitted with a power-law form. Since C₁ affects the index very weakly in this fitting, we just let C₁ = 0 in Figure 12. The height of the inner torus increases when its radius increases with the relation h ∝ R^0.69 ∝ L^0.25, which controls the geometry of torus.

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RM based on infrared emission is consistent with R_torus ∝ L^0.5_opt (Suganuma et al. 2006), which agrees with the prediction of a toy model for the sublimation process (Krolik & Begelman 1988). However, Suganuma et al. (2006) did not provide the index' error of R_torus–L_opt relation. We refit the data with error and the best fitting is R_torus ∝ L^0.46±0.09_opt, which is statistically consistent with our relation between IMLR radius and luminosity (R_IMLR ∝ L^0.37±0.06_opt).

However, the radius of the IMLR that we calculated here is strongly dependent on the radius of the VBLR, i.e., dependent on the R_BLR ∼ L5100 relation. If Equation (5) is used to calculate the radius of the BLR which is used as the radius of the VBLR here, we would obtain R_IMLR ∝ L^0.53±0.07, which is also consistent with the infrared RM result of R_torus ∝ L^0.46±0.09_opt. However, we prefer Equation (6) because it has taken into account the correction of starlight (Bentz et al. 2006). Equation (6) is also consistent with a simple radiation pressured dominated VBLR where L/R² = constant.

All these analyses suggest that the IMLR has roughly the same receding velocity as torus. Combined with the analysis in the sections above, we conclude that the IMLR is the inner part of a dust torus.

Based on the analysis above, a simple scenario of the BLR (BLR=VBLR+IMLR) evolution can be constructed as shown in Figure 13. The inner spherical region is the VBLR. It expands to a larger radius with luminosity increase and perhaps also black hole mass increase. We suggest that the IMLR is the inner part of the torus, which can be sublimated by the central radiation and thus its radius also increases with luminosity increase. Naturally, the IMLR will be photoionized by the irradiation of the central AGN and the material gravitationally bound by the central black hole will be virialized with the gravitational potential of the central black hole, as determined by Equations (1) and (2). Since the height of the torus increases slower than the inner radius of the torus when luminosity increases, the covering factor of the IMLR decreases with luminosity (and larger radius) as shown in the cartoon. When the luminosity is high enough, the two regions may eventually merge into one. The observed broad emission line is thus the superposition of the emission from the two regions. When the luminosity is large, the BLR and torus become one single entity, but with different physical conditions. This scenario is consistent with the dust bound BLR hypothesis (Laor 2004; Elitzur 2006). Suganuma et al. (2006) also showed that the delay time of infrared emission is always longer than the delay time of the emission lines of the corresponding AGN, but sometimes they are very close to each other and become almost the same. This is also consistent with the inner torus region as the origin of the IMGC.

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A piece of supportive evidence for the existence of the IMLR comes from the study of the microlensing of the BLR in the lensed quasar J1131–1231 (Sluse et al. 2007, 2008). In this work, they found evidence that the Hβ emission line (as well as Hα) is differentially microlensed, with the broadest component (FWHM 4000 km s⁻¹) being much more microlensed than the narrower component (FWHM 2000 km s⁻¹). The emission line can be well decomposed into two components and the emission line' profile has been significantly changed after microlensing compared to the original line. Because the amplitude of microlensing depends on the size of the emitting region, it can be naturally explained as the broadest component of the emission line that comes from a more compact region than that of the narrower component. Although it can also be explained as a single region with a range of gas velocities at different radii, the explanation that comes out from our model is natural.

Then, we turn to the RM experiment. If the IMLR is really a physically distinct emission region apart from the VBLR, another peak corresponding to the radius of this region might appear in the cross-correlation function between the light curves of the emission line and the continuum. In fact, a possible example exists in the database of RM observations (Peterson et al. 1998; 2004). Mrk 79 shows obviously double peaks in the cross correlation centroid distribution map as shown in Figure 14, which has not been explained well so far. There are four subsets of RM data for Mrk 79 which were taken in different time periods, and the double-peaked cross-correlation centroid distribution of Mrk 79 is from the fourth period. The measured time delays are about 9–16 days in the subsets 1–3 (Peterson et al. 2004), while in the fourth subset there are two typical time delays, one is about 6–10 days which is consistent with subsets 1–3, and the other is about 42 days, as shown in Figure 14. Here, we suggest that the shorter time delay, i.e., 6–10 days, is the delay time of the VBLR, which was also observed in subsets 1–3. The longer time delay, i.e., ∼42 days, is possibly the delay time of the IMLR, which was only observed in the fourth subset. The mean Hβ spectrum of Mrk 79 can be well described in a three Gaussian component model (Peterson et al. 1999), i.e., a normal narrow line, an IMGC and a VBGC, as shown in Figure 15. Based on the simple relation in Section 4.1 (Equation (3)), we get R_IMLR/R_VBLR = V²_VBLR/V²_IMLR = (5856 km s⁻¹/2522 km s⁻¹)² = 5.4. Therefore, the delay time for the IMLR should be 32–54 days, fully agrees with the second peak of CCCD. This agreement also suggests that our rather arbitrary choice of C₀ = f₁/f₂ = 1 does not deviate from its true value significantly, at least not for Mrk 79. However, it should be noted that in the mean Hβ the VBGC dominates over the IMGC, but in the CCCD the second peak appears to be much stronger. Probably, in the fourth subset, the IMLR responds to the continuum more than the VBLR, i.e., the IMGC is more variable.

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On the other hand, for the majority of other sources with RM observations, the IMGC is not important. We re-examined the 10 RM sources with public data on Peterson's Web site.⁶ Six of them only need one Gaussian component to describe the broad line of Hβ, one of them has double peak profile and one has irregular profile, only two of them need VBGC+IMGC to fit. In addition, 90% of the sources that plot on the R–L relation (Kaspi et al. 2005) have luminosity larger than 10⁴³ erg s⁻¹ and more than 60% have larger than 10⁴⁴ erg s⁻¹, which prefers the emission line to be one single Gaussian profile (in our sample, there are 14 sources' Hβ only need one Gaussian component, 11 of them have luminosity larger than 10⁴³ erg s⁻¹ and six of them have larger than 10⁴⁴ erg s⁻¹). Therefore, the R–L relationship established in emission line RM measurements may be valid only for the VBLR, i.e., the R–L relation only gives the radius of the VBLR, as assumed in Section 4.1. This is probably why we did not see double peaks in most of the CCCD maps (as shown above, even for Mrk 79, the two peaks were obtained in only one observation period out of the four periods in total). Two of the sources with double Gaussian line profiles are NGC 4051 and Mrk 509; the lag times for the two components of both sources satisfy this simple relationship t_IMLR/t_VBLR = V²_VBLR/V²_IMLR, consistent with our two-component BLR assumption. Interestingly, the lag time for the VBGC of NGC 4051 is between 1 and 2 days, significantly different from the lag time of about 4 days for the whole line. This revised lag time makes NGC 4051 consistent with the R–L relationship in Equation (5) or (6), resolving the outstanding problem of the significant deviation of NGC 4051 from the R–L relationship. The detailed results on our reanalysis of the RM data with this two-component model will be presented elsewhere (Zhu & Zhang 2009).

We propose that the VBLR and IMLR are identified with the variable regions that scales with luminosity. If this is the case, one might expect that the VBGC would vary more than the mean spectrum as it has smaller radius. However, that is not observed; the rms spectrum of the emission line is normally (but not always) narrower systematically than the mean spectrum (Collin et al. 2006). In our model, the IMLR has a flattened geometry, and our simulations show that it can create a narrower response function to a delta impulse even with radius much larger than the VBLR. We carried out a straightforward test by calculating the responses of the two components to the continuum. In this simple geometrical model, the VBLR is a spherical shell and the IMLR is a cylindrical shell (representing a flattened disk-like geometry with inner and outer boundaries). The gas density inside each region is independent of radius, i.e., the gas distribution could be clustered, but the distribution of gas clusters is independent of radius. The thickness of the VBLR shell is chosen to be three times that of the IMLR, because the VBLR is thought to have much lower density. The emissivity law of the two emission line regions is chosen in such a simple way that the emission line flux is proportional to the density times the continuum flux received at any point in the two regions. This means that the broad emission line intensity is proportional to the covering factor of each region. The line profile at any radius is assumed to have a Gaussian profile, with velocity determined from Equation (3). The overall broad emission line from each region is thus the superposition of all broad lines from all radii. The response functions with different combinations of parameters for the two shells and the inclination angle of the IMLR are shown in Figure 16. It can be seen that the IMLR normally can produce a narrower response function when the inclination angle is not very large. A narrower rms spectrum can be easily calculated based on the narrower response function of the IMLR. However, the FWHM difference in modeled rms and mean spectra is not as significant as the observed. The simulation is carried out with noiseless, evenly sampled data, so further work is needed to see if this model can explain the narrower rms spectrum quite well. In any case, the narrower rms is not in conflict with our model of the two distinct BLRs, with the outer region having a flattened geometry. The requirement for small inclination angles simply suggests that most of the reverberation mapped objects have small inclination angles, a generic property of AGNs with broad emission lines. Further, more extensive calculations on the detailed responses of the two BLRs to characteristic continuum light curves of AGNs and comparisons with data will be presented in a future work (Zhu & Zhang 2009).

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RM-based black hole mass is calculated by the virial equation

$$\begin{equation} M_{\rm { BH}}=\frac{R_{\rm { BLR}}}{G}f^2\rm {FWHM}_{\rm { H}\beta }^2, \end{equation} \tag{ 7 }$$

where FWHM_Hβ is the FWHM of the whole Hβ, which represents the virial velocity of the BLR. For an isotropic velocity distribution, as generally assumed, $f=\sqrt{3}/2$ (Onken et al. 2004). R_BLR is the BLR radius that can be calculated from Equation (5) or (6). As assumed in Section 4.1 and further discussed in Section 5.2, the radius obtained from RM may represent the radius of the VBLR actually. If we take this assumption, the FWHM of the VBGC, instead of the FWHM of the whole line, should be used to calculate the black hole mass. We therefore correct the black hole mass in this way,

$$\begin{equation} M_{\rm { BHb}}=\frac{R_{\rm { VBLR}}}{G}f^2\rm {FWHM}^2\left(\frac{\rm {FWHMb}}{\rm {FWHM}}\right)^2, \end{equation} \tag{ 8 }$$

where FWHMb is the FWHM of the VBGC, and R_VBLR is taken as the R_BLR in Equation (5) or (6). It gives a more significant mass correction for NLS1s than for BLS1s as shown in Figure 17(a). The correction factor is near unity when FWHM reaches about 5000 km s⁻¹. We use L5100 as an indicator for continuum luminosity, and a correction for accretion rate has also been shown in Figure 17(c). After such correction, NLS1s still have smaller black hole masses but normal accretion rate in units of the Eddington rate.

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Fifteen objects in our samples have velocity dispersion measurement data (sigma) in Shen et al. (2008), as shown in Figure 18, where comparisons are made between black hole masses measured here and that predicted by the currently used M–sigma relation M ∝ σ⁴ (Tremaine et al. 2002). It is obvious that the masses of all NSL1 (filled symbols in the upper panel of Figure 18) are well below, but become more close to, the predictions of the M–sigma relation before and after the correction, respectively. For other AGNs no significant changes to their masses are introduced by the correction process. As shown in the lower panel of Figure 18, after the correction, the median value of (log M_Hβ–log M_σ) is much closer to zero, and the dispersion is reduced from 0.76 to 0.50 dex, which is consistent with the black hole mass uncertainty of 0.5 dex in RM (Peterson 2006). The effect of the correction is obvious, albeit small number statistics due to the limited sample.

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It has been shown that the receding torus model with constant torus height fails to provide a good fit to the data of type II AGN fraction as a function luminosity, and a good fitting can be given when h increases slowly with luminosity with the relation h ∝ L^0.23 (Simpson 2005), in excellent agreement with that of the IMLR as we have shown in Section 4.4. Note that here we consider the change of covering factor to be totally because of the change in the opening angle during the calculation, following Wang et al. (2005). This agreement suggests that the receding of torus is sufficient to explain the decrease of covering factor with decreasing luminosity. Our model is consistent with the torus that Simpson (2005) needed to explain underpopulated luminous type II AGNs.