Rarefied Broad-Line Regions in Active Galactic Nuclei:
Anomalous Responses in Reverberation Mapping and Implications for Weak Emission-Line Quasars

In the UV and optical spectra of active galactic nuclei (AGNs), strong broad emission lines (BELs) with velocity widths of $10^{3}$ – $10^{4}$ km/s are one of the most prominent features. BELs originate from the photoionization of the gaseous clouds in the broad-line regions (BLRs) driven by the continuum radiation from the accretion disks around the central supermassive black holes (SMBHs). The fluxes or equivalent widths (EWs) of BELs are determined by, e.g., the strength of ionizing radiation, the amount of BLR gas, or the reprocessing coefficients, while their line profiles are controlled by the bulk motions of BLR gas governed by the gravitational potential of SMBHs. Understanding the BELs and the underlying BLR physics is crucial to revealing the origin and evolution of AGNs.

Reverberation mapping (RM) is a classic tool for the investigation of BLR properties in AGNs and the mass measurement of their SMBHs (Bahcall et al., 1972; Blandford & McKee, 1982). It measures the time delays of emission lines (emission-line light curves) relative to the variation of the continuum (continuum light curve), and has been applied to more than a hundred of AGNs in the past three decades (e.g., Peterson et al., 1998; Kaspi et al., 2000, 2021; Bentz et al., 2009; Du et al., 2014, 2018; Grier et al., 2017; Lira et al., 2018; Yu et al., 2021). RM can probe the photoionization properties of the material in BLRs based on the response of emission-line flux with respect to the varying continuum (e.g., Goad et al., 1993; Gilbert & Peterson, 2003; Korista & Goad, 2004; Cackett & Horne, 2006) and diagnose the BLR geometry and kinematics from the response as a function of velocity (e.g., Welsh & Horne, 1991; Bentz et al., 2009; Denney et al., 2010; Grier et al., 2013; Pancoast et al., 2014; Du et al., 2016; U et al., 2022; Villafaña et al., 2022).

The theoretical calculations of RM signals based on photoionization models started in 1990s. For example, Goad et al. (1993) introduced photoionization models into the calculations of the responses for different emission lines for the first time. Bottorff et al. (1997) presented a more sophisticated kinematic model incorporating photoionization calculations for the one- and two-dimensional transfer functions of NGC 5548 (focusing on C iv emission line). Kaspi & Netzer (1999) performed photoionization calculations using a pressure-confined model in order to reproduce the light curves of five emission lines in the RM observations of NGC 5548. Korista & Goad (2000) adopted the locally optimally emitting clouds model (Baldwin et al., 1995) and obtained a better fitting to the UV emission-line light curves of NGC 5548. Negrete et al. (2014) used CLOUDY and the flux ratios of UV lines to derive the BLR radii. Goad & Korista (2014) investigated the influences to the variation amplitudes and time delays of emission lines from light-curve durations, sampling rates, and the time scales of driving continuum variabilities based on photoionization simulations. More recently, Guo et al. (2020) adopted photoionization models to explain the behavior of Mg II emission line in RM. Zhang et al. (2021) compared the BLR sizes measured from RM and spectroastrometry based on photoionization models. However, all of those calculations payed attention only to the BLRs of typical AGNs (with typical emission-line EWs, e.g., NGC 5548).

The EWs of BELs in AGNs have wide distributions, which roughly span more than one order of magnitude for the primary emission lines like H$\beta$, Mg ii, and C iv (e.g., Boroson & Green, 1992; Marziani et al., 2003; Shen et al., 2011). It means that the BLR properties, especially the gas content or ionization state, may be different for different objects. Furthermore, there are non-negligible populations of AGNs in which even weaker or more rarefied BLRs may reside. One population is the so-called “weak emission-line quasars (WLQs)” in high redshifts discovered in the past 20 years (e.g., Fan et al., 1999; Anderson et al., 2001; Collinge et al., 2005; Diamond-Stanic et al., 2009; Plotkin et al., 2010; Wu et al., 2011; Meusinger & Balafkan, 2014; Andika et al., 2020). They are characterized by the significantly weaker Ly$\alpha$ $\lambda 1216$ $+$ N v$\lambda 1240$ and/or C iv$\lambda 1549$ emission lines in the UV spectra (in rest frames) than the main population of quasars (e.g., Diamond-Stanic et al., 2009). They have high accretion rates but very weak BELs. There are also some other AGN populations at low redshifts which have relatively weak BELs, e.g., Seyfert 1.8/1.9 galaxies (e.g., Trippe et al., 2010), “naked” AGNs (e.g., Panessa et al., 2009). From theoretical perspective, the AGNs with relatively weak BELs (hosting rarefied BLRs) may be at the early stage of BLR evolution (Hryniewicz et al., 2010; Wang et al., 2012), which are probably important to understanding the AGN physics. However, the RM behaviors of the AGNs with rarefied BLRs (refer to low BLR gas densities) have not been investigated from theoretical calculation or systematically from observation. Recent RM observations of UV emission lines have revealed some anomalous behaviors (Lira et al., 2018). What’s surprising is that they just appear in the objects with weak emission-line EWs (CT320, CT803, and J224743 in Lira et al., 2018, see its Section 3.3). These UV lines show inverse correlations (negative responses) with the continuum variations (the emission-line flux goes down/up when the continuum flux increases/decreases, see more details in Section 5.1). This phenomenon is different from the “BLR holiday” anomaly in the emission lines of NGC 5548 (Goad et al., 2016; Pei et al., 2017) and Mrk 817 (Kara et al., 2021) found by the first and second AGN Space Telescope & Optical Reverberation Mapping (STORM) programs. During the “BLR holiday” period in AGN STORM, the emission lines decoupled from the continuum variations and showed weak (even no) correlations, which is different from the inverse correlations found in Lira et al. (2018) and can be explained by the obscuration from the disk wind (e.g., Dehghanian et al., 2019). These thoughts and current situations motivate us to investigate the RM behaviors of the emission lines in the AGNs with rarefied BLRs from photoionization calculations.

Furthermore, the scatter of the famous radius-luminosity ($R$-$L$) relationship discovered by RM (e.g., Kaspi et al., 2000; Bentz et al., 2013; Du & Wang, 2019) is also far from fully understood. Recent RM campaigns found that the scatter became larger with more objects (with different properties) been observed. For example, super-Eddington accreting massive black holes project found that the H$\beta$ lags of the AGNs with high accretion rates are shorter than the prediction of the $R$-$L$ relationship by factors of $\sim$3-8 (Du et al., 2015, 2016, 2018). Mg ii (Martínez-Aldama et al., 2020) and C iv (Dalla Bontà et al., 2020) emission lines also show preliminary signs of the similar behaviors. The Sloan Digital Sky Survey RM project discovered that the H$\beta$ lines of some quasars with moderate Eddington ratios also show shortened lags (Grier et al., 2017). The possible explanations for these shortened lags include (1) the self-shadowing effects of slim accretion disks in super-Eddington AGNs (Wang et al., 2014a) and (2) the variation of the spectral energy distribution caused by the spin of BHs (Wang et al., 2014b). However, it’s not yet known whether they are the only drivers for the scatter of the $R$-$L$ relationship. The influence of the BLR densities to the time lags has not been investigated systematically for different emission lines. This is also a main goal of this paper.

More recently, the RM observations of some objects showed a surprising phenomenon that their time lags changed significantly in different years, however the corresponding continuum luminosities were quite similar, e.g., NGC 3227 in Denney et al. (2010), De Rosa et al. (2018), and Brotherton et al. (2020), Mrk 817 in Peterson et al. (1998), Denney et al. (2010), and Lu et al. (2021), Mrk 79 in Lu et al. (2019) and Brotherton et al. (2020), and PG 0947+396 in different years in Bao et al. (2022). Considering that the time lags are mainly determined by the ionizing continuum and the properties of BLR gas, these observations indicate that their BLR might probably change with time because their continuum fluxes kept almost the same. The density of gas is one of the key BLR properties, and is therefore worthy of investigation.

This paper is organized as follows. The photoionization calculation is described in Section 2. A comparison between the EWs obtained from the photoionization models and the observations from large quasar samples or RM samples are provided in Section 3 for different emission lines. Section 4 presents the transfer functions for different BLR density distributions, as well as the correlations between EWs and time lags. In this section, we demonstrate that the AGNs with rarefied BLRs may show anomalous RM behaviors. Some discussions are provided in Section 5 (especially the implications for WLQs). Finally, we briefly summarize in Section 6.

Following the pioneering works of photoionization calculations for the emission-line responses in RM (e.g., Korista & Goad, 2000, 2004), we adopt the locally optimally emitting cloud (LOC, Baldwin et al., 1995) model and perform the calculation using CLOUDY v17.02 (Ferland et al., 2017). We generate a grid of models with gas number density of $7\leqslant\log\ (n_{\rm H}/{\rm cm^{-3}})\leqslant 15$ and surface flux of ionizing photons spanning $16\leqslant\log\ (\Phi_{\rm H}/{\rm cm^{-2}\,s^{-1}})\leqslant 24$. The steps in $n_{\rm H}$ and $\Phi_{\rm H}$ are both 0.125 dex. We assume a simple slab geometry with column density of $10^{23}\ {\rm cm^{-2}}$ (a standard value, see Netzer & Marziani 2010 and references therein) for the line-emitting entities. The typical spectral energy distribution (SED) of Mathews & Ferland (1987) is employed as the ionizing continuum (more discussion is given in Section 5.4). The metallicity is assumed to be solar abundance.

We focus on the emission lines that frequently presented in the studies of UV (e.g., Clavel et al., 1991; Grier et al., 2019; Kaspi et al., 2021; Lira et al., 2018; Yu et al., 2021) and optical RM observations (e.g., Peterson et al., 1998; Kaspi et al., 2000; Du et al., 2014; Grier et al., 2017; U et al., 2022). They are Ly$\alpha$ $\lambda 1216$, Si iv$+$O iv] $\lambda 1400$ blend, C iv $\lambda 1549$ doublet, C iii] $\lambda 1909$ blend, Mg ii $\lambda 2798$ doublet, and H$\beta$ $\lambda 4861$ emission lines. There are many other emission lines (Al iii $\lambda\lambda 1855,1863$, Si iii $\lambda\lambda 1883,1892$, and Fe iii $\lambda\lambda 1895,1914,1926$) that are seriously blended with C iii] (see, e.g., Negrete et al., 2012; Temple et al., 2020). They are also added into the flux of C iii] blend.

In order to demonstrate the responses of the BLR clouds to variation of the continuum for different $\Phi_{\rm H}$ and $n_{\rm H}$, we adopt the definition of responsivity ($\eta$) in Korista & Goad (2004). It is defined as

The responsivities of the BLR clouds with different $n_{\rm H}$ and $\Phi_{\rm H}$ obtained from the photoionization grid, as well as the corresponding EWs and fluxes, are shown in Figure 1. In each panel of the responsivity in Figure 1, a solid line is added as the division of positive and negative values. It is obvious that the responsivity is negative if the gas density is low and the surface flux of the ionizing photons is high (in the upper-left corners of the corresponding panels in Figure 1). This motivates us to consider that the emission-line responses may gradually become negative if the weights of low-density clouds in BLRs increase.

Integrating the line emission over the grid, we can obtain the luminosities of the emission lines. Following Baldwin et al. (1995) and Bottorff et al. (2002), the emission-line luminosity can be obtained by

where $F_{\ell}(n,r)$ is the line flux of the clouds with gas density $n$ at radius $r$, $f(r)$ is the covering fraction, and $g(n)$ is the gas density distribution. If the ionizing luminosity $L_{\rm ion}$ is given, the radius and the flux of ionizing photons are connected through $\Phi_{\rm H}\propto L_{\rm ion}/4\pi r^{2}$. We assume that both of $f(r)$ and $g(n)$ are simply power laws, namely $f(r)\propto r^{\Gamma}$ and $g(n)\propto n^{\beta}$, where $\Gamma$ and $\beta$ are two indexes. The goal of the present paper is to investigate the potential observational characteristics in RM if the BLR densities change. We fix $\Gamma$ as a constant and keep $\beta$ as a free parameter. As an AGN prototype, the optimal $\Gamma$ in NGC 5548 was suggested to be in the range of $-1.4<\Gamma<-1.0$ (Korista & Goad, 2004). For simplicity, we adopt $\Gamma=-1.0$ in this work.

For typical AGNs, $\beta=-1.0$ is an acceptable assumption (Baldwin et al., 1995; Baldwin, 1997; Korista & Goad, 2004; Guo et al., 2020). Smaller $\beta$ means that most of the BLR clouds have lower gas densities (rarefied BLRs), and larger $\beta$ stands for the BLRs with higher densities (dense BLRs). Here we set $\beta$ from $-3.0$ to $0.0$ to check the influences of gas density to the response behaviors of different emission lines in RM observations. For UV emission lines, we adopt $n_{\rm min}=10^{8}\,{\rm cm^{-3}}$ and $n_{\rm max}=10^{12}\,{\rm cm^{-3}}$ because lower density may lead to some unobserved forbidden lines and higher density makes, e.g., C iv thermalized (Korista & Goad, 2000, 2004). However, hydrogen recombination line can emit efficiently even in much higher density (see Figure 1 and Korista et al. 1997). We loose $n_{\rm max}$ to $10^{15}\,{\rm cm^{-3}}$ for the calculation of the H$\beta$ emission line.

To obtain the emission-line luminosity (Eqn 2), the inner and outer radii ($r_{\rm in}$ and $r_{\rm out}$) of BLRs are required. From Figure 1, the most efficiently emitting clouds are located in relatively a narrow range of $\Phi_{\rm H}$. In the present paper, we pay more attention to the influences of different Eddington ratios than those of the BH masses. For a given BH mass $M_{\bullet}$, we adopt $r_{\rm in}=\sqrt{Q_{0.1}/4\pi\Phi_{\rm H}^{\rm max}}$ and $r_{\rm out}=\sqrt{Q_{0.1}/4\pi\Phi_{\rm H}^{\rm min}}$, where $\Phi_{\rm H}^{\rm min}=10^{18}$ cm^-2 s^-1 and $\Phi_{\rm H}^{\rm max}=10^{22}$ cm^-2 s^-1. $Q_{0.1}$ is the total number of ionizing photons if the Eddington ratio $L_{\rm Bol}/L_{\rm Edd}=0.1$, where $L_{\rm Bol}=\kappa L_{5100}$ is the bolometric luminosity, $\kappa$ is the bolometric correction factor, and $L_{\rm Edd}=1.26\times 10^{38}M_{\bullet}\,{\rm erg\,s^{-1}}$ is the Eddington luminosity. We simply adopt $\kappa=10$ here (e.g., Richards et al., 2006), but caution that $\kappa$ may depend on the accretion rate or SMBH mass (e.g., Jin et al., 2012). The $\Phi_{\rm H}^{\rm min}$ and $\Phi_{\rm H}^{\rm max}$ values we adopted here are just corresponding to $r_{\rm in}=0.1r_{\rm BLR}^{\rm typical}$ and $r_{\rm out}=10r_{\rm BLR}^{\rm typical}$, where $r_{\rm BLR}^{\rm typical}$ is the typical BLR size of the AGNs with $L_{\rm Bol}/L_{\rm Edd}=0.1$ calculated from the classical radius-luminosity (R-L) relationship of $\log(r_{\rm BLR}/{\rm lt\!-\!days})=1.53+0.53\log\ell_{44}$ in Bentz et al. (2013), where $\ell_{44}=L_{5100}/10^{44}\ {\rm erg\ s^{-1}}$ is the monochromatic luminosity at 5100Å. $L_{\rm Bol}/L_{\rm Edd}=0.1$ is roughly the central value of Eddington ratios for nearby Seyfert galaxies and high-redshift quasars ($0.01\lesssim L_{\rm Bol}/L_{\rm Edd}\lesssim 1$, e.g., Boroson & Green 1992; Marziani et al. 2003; Shen et al. 2011; Du & Wang 2019; Wu et al. 2015, and see Figure 4), so the corresponding $r_{\rm BLR}^{\rm typical}$ can be regarded as a typical value. It should be noted that $L_{\rm Bol}/L_{\rm Edd}=0.1$ here is only used for determining a relatively reasonable $r_{\rm in}$ and $r_{\rm out}$. For higher (or lower) ionizing luminosity $L_{\rm ion}$ (or Eddington ratio $L_{\rm Bol}/L_{\rm Edd}$), the most efficiently emitting radius of the line region will spontaneously increases (or decreases) in accordance with $r\propto L_{\rm ion}^{1/2}$. This is known as the physical interpretation for the R-L relationship established from the RM campaigns (e.g., Kaspi et al., 2000; Bentz et al., 2013). Such boundary assumption makes the EW calculation scale-free from BH mass. The dynamic range of radius adopted here is generally large enough for the span of Eddington ratio. We have checked that slightly larger or smaller $r_{\rm in}$ and $r_{\rm out}$ do not influence the general results of the present paper.

In Figure 4, we show the dependences of EW on $L_{\rm Bol}/L_{\rm Edd}$ for the emission lines obtained from our photoionization model. The cases with overall covering factors of $C_{\rm f}=20\%$ and $70\%$ are demonstrated. In general, smaller $\beta$ (more rarefied BLR) tends to show weaker EWs. But at the low Eddington-ratio ends of Si iv$+$O iv], C iv, and C iii], large $\beta$ (dense BLR) also produces small EWs.

In order to compare our theoretical calculations with observations and to generally determine the ranges of $\beta$ for different emission lines, we collect the emission-line EWs and $L_{\rm Bol}/L_{\rm Edd}$ from the following samples: (1) the objects from recent RM campaigns or compilations (Grier et al., 2017; Lira et al., 2018; Du & Wang, 2019; Homayouni et al., 2020; Yu et al., 2021; Kaspi et al., 2021), (2) the H$\beta$ (Boroson & Green, 1992) and UV (Kuraszkiewicz et al., 2002, 2004) emission lines of PG quasars, (3) the emission lines of the quasar samples (Calderone et al., 2017; Rakshit et al., 2020) from Sloan Digital Sky Survey (SDSS), and (4) the WLQs from Shemmer et al. (2010), Wu et al. (2011), and Plotkin et al. (2015).

For the RM objects, we can easily calculate their Eddington ratios based on the BH masses obtained from the time lags (Grier et al., 2017; Lira et al., 2018; Du & Wang, 2019; Homayouni et al., 2020; Yu et al., 2021; Kaspi et al., 2021). All of the objects in PG sample have H$\beta$ observations. For the PG quasars without RM measurements, the single-epoch BH masses estimated from their H$\beta$ lines can be used. We employ the simple virial relation to determine their BH masses, namely

where $V_{\rm H\beta}$ is the FWHM of H$\beta$ line, $G$ is the gravitational constant, and $f$ is the virial factor. We simply adopt $f=1$ in this paper (e.g., Woo et al., 2015). As aforementioned, the recent works (e.g., Du et al., 2015, 2018; Du & Wang, 2019) discovered that BLR radius (measured from H$\beta$) depends on accretion rate and suggested a new scaling relation for $r_{\rm H\beta}$, which includes the relative strength of Fe ii lines as a new parameter. We calculate the BH masses of the PG objects, which have both H$\beta$ and Fe ii measurements, using the new scaling relation of

in Du & Wang (2019), where $R_{\rm Fe}$ is the flux ratio between Fe ii and H$\beta$ lines. For the SDSS quasars with low redshifts which have H$\beta$ observations, the single-epoch BH masses based on the new scaling relation are also used. For the SDSS quasars with high redshifts (with only UV lines), their BH masses and Eddington ratios are based on the classic single-epoch BH mass estimators (see Calderone et al., 2017; Rakshit et al., 2020). It should be noted that the Eddington ratios of the SDSS quasars with high redshifts (for the UV lines) may be underestimated to some extent because their BH masses are obtained based on the classic single-epoch BH mass estimators and the shortening effects of the time lags (e.g., Martínez-Aldama et al., 2020; Dalla Bontà et al., 2020) have not been taken into account. For the WLQs, we only select the objects that have both of C iv and H$\beta$ measurements (Shemmer et al., 2010; Wu et al., 2011; Plotkin et al., 2015). An advantage of selecting these WLQs is that, we can obtain relatively good $M_{\bullet}$ estimates for these objects using the new scaling relation (Du & Wang, 2019), considering that the H$\beta$ emission lines in WLQs are not significantly different from those of normal AGNs (see more details in Section 5.3).

In Figure 4, the emission lines show the Baldwin effects (e.g., Baldwin, 1977) but with different slopes and significances. In general, our photoionization models can cover the distributions of the observational points, which validates the settings of the parameters used in our calculations. Comparing the EW distributions of the current RM samples with those of the other samples, it is obvious that there is still room to improve the completeness of the RM samples. For example, the current RM sample of C iv lacks low-EW objects, especially WLQs, and the H$\beta$ RM sample lines also shows a little bias toward high EWs. On the contrary, more RM observations of Ly$\alpha$, Si iv$+$O iv], and C III] emission lines are needed for high-EW AGNs. Moreover, it is obvious that the objects, which showed anomalous RM behaviors in Lira et al. (2018) (CT320, CT803, and 2QZJ224743), are located at the lower EW ends of the corresponding panels in Figure 4. It implies that their anomalous behaviors may probably connect with weak EWs.

Figure 4 can give a general constraints to the parameter $\beta$ in the context of the photoionization calculations in the present paper. For example, we can determine that the $\beta$ parameter of WLQs roughly ranges from $-2.0$ to $-1.0$ for C iv and from $-0.7$ to $0.0$ for H$\beta$ for the case with $C_{\rm f}=20\%$, and from $-2.5$ to $-1.5$ for C iv and from $-1.2$ to $-0.5$ for H$\beta$ for the case with $C_{\rm f}=70\%$, respectively. It is noted that the observational distributions of the C III] blend in Figure 4 depart slightly from the photoionization calculations. One probable reason is the relatively low abundance (solar abundance) we adopted here (e.g., Snedden & Gaskell 1999, also see Panda et al. 2018, 2019; Śniegowska et al. 2021). We mainly focus on the influence of the BLR densities to the RM behaviors. The influence from abundance will be discussed in a future separate paper (see also Section 5.4 and Appendix A).

In RM, the delayed response of an emission line to the varying continuum can be characterized by the transfer function (Blandford & McKee, 1982) in the form of

where $\Delta F_{\rm c}(t)$ and $\Delta F_{\ell}(t)$ are the variations of the continuum and emission-line fluxes, and $\Psi(t)$ is the one-dimensional transfer function. Transfer function $\Psi$ connects the emission-line light curve with the continuum variation (Blandford & McKee, 1982), and can be easily obtained from RM observations by different algorithms and softwares, e.g., the maximum entropy method (e.g., Krolik et al., 1991; Horne et al., 1991), JAVELIN (Zu et al., 2011), MICA (Li et al., 2016), and Pixon (Li et al., 2021). Some examples of the one-dimensional transfer functions reconstructed from RM observations can be found in, e.g., Grier et al. (2013), Williams et al. (2018), and Bao et al. (2022). From the photoionization grid in Section 2, we can calculate the one-dimensional transfer functions by

where

is the emission-line flux at radius $r$ obtained by integrating $F(n,r)$ over density $n$,

is the responsivity function at radius $r$ derived from Eqn (1), $\tau$ is the time, $\vec{n}_{\rm obs}$ is the line of sight, $c$ is the speed of light, and ($\theta$, $\phi$) are the angles of the spherical coordinates. The purpose of this paper is to investigate the influence of BLR densities to the transfer functions. Therefore, for simplicity, we assume that the BLR geometry is spherically symmetric.

We calculate the transfer functions for different $\beta$ and show the results in Figure 5 for three cases of different Eddington ratios [$\log(L_{\rm Bol}/L_{\rm Edd})=-1.5$, $-0.5$, and $0.5$]. The mass of SMBH is set to be $10^{8}M_{\odot}$. Generally speaking, at low Eddington ratios, the transfer functions are always positive and their peaks (with the strongest responses) move toward longer time lags if $\beta$ decreases (more rarefied BLRs). Along with Eddington ratio increases, the transfer functions with smaller $\beta$ (more rarefied BLRs) still have longer time lags than those with larger $\beta$ (denser ones), however their amplitudes become from positive to negative. Negative transfer functions mean that the emission-line light curves show inverse responses with respect to the variations of the continuum light curves, which are different from the usual cases in RM. To quantify whether or not the response is negative in average, we calculate the average of a transfer function by $\bar{\Psi}=\int\Psi(t)dt/\int dt$. $\bar{\Psi}$ for different $\beta$ are also shown in Figure 5. The transition from positive $\bar{\Psi}$ to negative $\bar{\Psi}$ happens at larger $\beta$ if Eddington ratio increases. More specifically, different lines show different behaviors. At the same Eddington ratio, it is easier for the responses of Mg ii, C iv, and C iii] to become negative than those of Ly$\alpha$, Si iv$+$O iv], and H$\beta$. The transitions of Mg ii, C iv, and C iii] occur at relatively larger $\beta$. For example, Mg ii can become negative at $\log(L_{\rm Bol}/L_{\rm Edd})=-0.5$, however the average responses of H$\beta$ keep positive in the same case.

The $\beta$ parameter controls both of EWs and typical BLR radii (time lags), therefore we can investigate their correlations. We define an average time lag of a transfer function using the formula of

Through adjusting $\beta$, we can show the correlations between time lags and EWs in Figure 8. It should be noted that this definition is not appropriate if $\Psi(t)$ is partly negative. Considering that currently only positive transfer functions are thought to be “successful” in RM observations, we plot the cases of purely positive transfer functions here. Therefore, the curves for high Eddington ratios in Figure 8 are cut off at small $\beta$ (see, e.g., the cases of C iv for $\log(L_{\rm Bol}/L_{\rm Edd})=0.1$ and $0.5$ in Figure 8).

In Figure 8, each line has the same luminosity or Eddington ratio, and is color-coded by the parameter $\beta$. These correlations mean that the diverse density distributions of BLRs can result in variations of EWs (by nearly an order of magnitude) and time lags (by factors of $\sim$2-3) simultaneously even if the luminosities are the same. This may contribute to the scatters ($\sim$0.3dex) of the $R$-$L$ relationships of the emission lines and may also be one of the explanations for the variations of time lags in individual objects in different years. However, the correlations between time lags and EWs are different in different emission lines. Ly$\alpha$, Si iv$+$O iv], C iv, and C iii] show nonmonotonic correlations. In these four emission lines, along with $\beta$ increasing, the time lags continuously become shorter, however the EWs first increase and then decrease after across their maximums. For the other two lines (Mg ii and H$\beta$), both of the time lags and EWs show almost monotonic correlations with $\beta$. $\beta$ shows correlations (and anti-correlations) with their EWs (and time lags). More specifically, for the three primary emission lines in RM - C iv, Mg ii, and H$\beta$, they have the following typical characteristics.

C iv: The EWs and time lags mainly show positive correlations at low Eddington ratios [e.g., $\log(L_{\rm Bol}/L_{\rm Edd})=-2.5$ and $-2.1$] and anti-correlations at high Eddington ratios [e.g., $\log(L_{\rm Bol}/L_{\rm Edd})=0.1$ and $0.5$]. The change of BLR density ($\beta$) causes the variation of time lag but only weakly influence the EWs at moderate Eddington ratios [e.g., $\log(L_{\rm Bol}/L_{\rm Edd})=-0.6$].

Mg ii: The EWs and time lags are mainly anti-correlated. Time lag decreases if EW increases at almost all Eddington ratios. However, the slopes are steeper at low EWs than those at high EWs.

H$\beta$: The behavior of H$\beta$ line is more similar to that of Mg ii line. However, at high EWs (e.g., ${\rm EW}\gtrsim 80$-$100$ for the cases of $C_{\rm f}=70\%$ and ${\rm EW}\gtrsim 30$-$40$ for the cases of $C_{\rm f}=20\%$), the variations of the time lags are very weak along with the change of $\beta$.

Here we do not plot the RM observations in Figure 8 because the exact values of time lags are also significantly controlled by BLR geometry, which is simply assumed to be spherically symmetric in our calculations (may be very different from the actual situations). But we can generally do a simple comparison. Du & Wang (2019) shows the correlation between $\Delta R_{\rm H\beta}=\log\,(R_{\rm H\beta}/R_{\rm H\beta,R-L})$ and ${\rm EW_{H\beta}}$, where $R_{\rm H\beta,R-L}$ is the H$\beta$ lag calculated from the $R$-$L$ relationship. Each line in Figure 8 has the same luminosity. Therefore, we can compare the observed $\Delta R_{\rm H\beta}$-${\rm EW_{H\beta}}$ correlation with our calculations. In Du & Wang (2019), except for those super-Eddington AGNs, the AGNs with normal accretion rates (with $70<{\rm EW_{H\beta}}<300$) do not show any significant $\Delta R_{\rm H\beta}$-${\rm EW_{H\beta}}$ correlation or anti-correlation. This is generally consistent with our calculations that ${\rm EW_{H\beta}}$ and $\tau_{\rm H\beta}$ do not show strong correlation at relatively high EWs. From our calculations, we expect that more high-quality observations of the H$\beta$ with weaker EWs in AGNs with normal accretion rates in future may discover some objects with longer time lags than the $R$-$L$ relationship.

In addition, we simply check the RM samples of C iv lines (Lira et al., 2018; Grier et al., 2019; Kaspi et al., 2021) and obtain a very weakly positive $\Delta R_{\rm CIV}$-${\rm EW_{CIV}}$ correlation with Spearman’s rank correlation coefficient of $\rho=0.32$ and a corresponding $p$-value of $0.09$ based on the C iv $R$-$L$ relationship in Kaspi et al. (2021). Considering that the $\beta$ values of the current C iv RM samples are relatively large (see Figure 4), this may probably be consistent with our calculations, especially the parts with $\tau<200$ days in Figure 8. Similarly, we also perform a test to the Mg ii samples (Lira et al., 2018; Homayouni et al., 2020; Yu et al., 2021) based on the $R$-$L$ relationship in Homayouni et al. (2020). However, no significant correlation is found ($\rho=0.09$ and $p=0.61$). One possible reason is the relatively narrow EW span (0.16dex) of the current Mg ii RM samples (Lira et al., 2018; Homayouni et al., 2020; Yu et al., 2021). As a comparison, the EW span of C iv RM samples (Lira et al., 2018; Grier et al., 2019; Kaspi et al., 2021) is 0.30dex.

Through adjusting $\beta$, we can also plot the correlation between $\bar{\Psi}$ and EW in Figure 9 in order to check the EW at which the transition from averagely positive to negative response happens. From Figure 9, C iv, Ly$\alpha$ are more easier to show negative (anomalous) responses in RM because the grey ranges cover more cases with negative $\bar{\Psi}$. The observations of the other four lines can also cover the negative responses from the photoionization calculations, however only if the EWs are very close to the lower limits.

In this paper, we present photoionization calculations (LOC models) for the one-dimensional transfer functions of Ly$\alpha$ $\lambda 1216$, Si iv$+$O iv] $\lambda 1400$ blend, C iv $\lambda 1549$ doublet, C iii] $\lambda 1909$ blend, Mg ii $\lambda 2798$ doublet, and H$\beta$ $\lambda 4861$ emission lines in order to investigate the roles of BLR densities in their RM observations. Based on the calculations and the comparison with observations, we have the following predictions and conclusions:

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  The AGNs with rarefied BLRs (small $\beta$) are predicted to show negative responses (anomalous responses) in RM observations. The emission lines of such objects may have relatively low EWs. The observed anomalous behaviors in the UV emission lines of some objects in the past RM campaigns (e.g., CT 320, CT 803, and 2QZ J224743 in Lira et al. 2018) may be explained by the rarefied BLRs. In this case, the emission-line light curves may need to be flipped before the time-series analysis if we want to get accurate BLR radii.

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  The different BLR densities in AGNs may contribute to the scatter of the $R$-$L$ relationship. The wide distributions of the BLR densities can result in changes of time lags by factors of 2-3. Preliminarily, the observed scatter of the C iv $R$-$L$ relationship is probably consistent with the calculations in the present paper. For the other emission lines (Mg ii and H$\beta$), more RM observations for the AGNs with wider EW spans are needed.

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  The variation of time lags in individual objects without significant changes of continuum luminosities may be explained by the changes of BLR densities with time.

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  We propose that the existence of negative responses in C iv RM observations can be used to justify if the anemic BLR model works or not in WLQs. If negative responses are found in WLQs, their emission-line weakness can be attributed to the deficit of BLR gas.