Interpretation of Departure from the Broad-line Region Scaling in Active Galactic Nuclei

Since Wang et al. (2014a) showed that, most likely, a high value of spin is required, in the present paper we use the Novikov–Thorne model of the accretion disk (Novikov & Thorne 1973; Wang et al. 2014a). We combine the local emissivity with the ray-tracing procedure applied earlier in Czerny et al. (2011), which allows us to include all relativistic corrections, including the light bending, gravitational redshift, and Doppler boosting. The disk model is thus parameterized by the black hole mass, M_•, accretion rate, ${\dot{M}}_{\bullet }$, and dimensionless spin parameter, a. We consider both prograde and retrograde spin values.

The calculation of the observed monochromatic luminosity in this model depends on the viewing angle, i, between the symmetry axis and the observer. We do not know this angle in individual sources, but the expected range of angles is constrained by the presence of the dusty molecular torus (see Krolik 1999). The torus blocks the view toward the nucleus at high viewing angles; such sources do not show their BLR, and they are classified as type 2 sources. Observational constraints on the torus opening angle are not simple; they typically imply a mean torus opening angle of order of 45^◦, and this value does not depend strongly on the source luminosity (e.g., Tovmassian 2001; Lawrence & Elvis 2010; He et al. 2018). Taking into account this constraint and assuming otherwise random orientation of AGNs with respect to us, we adopt viewing angle i ∼ 30° as a representative value. Thus, we measure the observed fluxes assuming the same viewing angle for all the sources. We neglect here the effect of finite wavelength width for the hydrogen cross section (see Equation (5) of Wang et al. 2014a) and determine the luminosity at 13.6 eV (1 ryd) as

$$\begin{eqnarray}&&{L}_{\mathrm{ion}}=\nu {L}_{\nu }(1\,\mathrm{Rydberg})\,[\mathrm{erg}\,{{\rm{s}}}^{-1}].\end{eqnarray} \tag{ 1 }$$

However, the BLR sees a different part of the spectrum. The BLR covers from 10% to 30% of the sky from the point of view of the inner disk, and it intercepts photons propagating relatively close to the equatorial plane (but not too close since highly inclined photons are intercepted by the disk itself). So, when calculating the number of ionizing photons, we integrate the spectrum above 1 ryd over all viewing angles between 80° and 45°:

$$\begin{eqnarray}&&Q={\int }_{{45}^{^\circ }}^{{80}^{^\circ }}{\int }_{1\,\mathrm{Rydberg}}^{\infty }\displaystyle \frac{{L}_{\nu }(\theta )}{h\nu }d\nu \sin \theta d\theta \,[\mathrm{photons}\,{{\rm{s}}}^{-1}].\end{eqnarray} \tag{ 2 }$$

The integration is performed far from the black hole, where the general relativity effects are already negligible. Photons propagating at angles larger than 80^◦ are neglected since they will be absorbed by the disk itself. The ionization level of the BLR clouds can be estimated using either L_ion or Q.

The transition from the outer cold disk to the inner hot disk is still debated, particularly for the galactic sources. However, it is clear that in very low luminosity sources (the most extreme case is Sgr A*) there is no cold outer disk. A series of papers discussed this issue on the basis of the radiative and conductive interaction between the hot corona above the disk and the underlying cold disk, and it was shown that for very low accretion rates the cold inner disk disappears (see Yuan & Narayan 2014, for a review). The inner flow then proceeds through an optically thin hot flow, such as ADAF (Ichimaru 1977; Narayan & Yi 1994) or its alternatives, e.g., advection-dominated inflow–outflow solutions (ADIOS; Blandford & Begelman 1999). Most of these solutions can be described under the common name RIAF (radiatively inefficient accretion flow), but some are actually quite radiatively efficient if the strong coupling exists between the hot ions and electrons (Bisnovatyi-Kogan & Lovelace 1997; Sironi & Narayan 2015). However, the common property of these solutions is that ion temperature is close to virial temperature, and electrons are also relatively hot, so the emitted radiation concentrates in X-rays instead of far-UV local blackbody emission characteristic for the cold accretion disks.

For the purpose of this paper we use two prescriptions from Czerny et al. (2004). The first one is simply based on the strong ADAF principle, i.e., whenever the ADAF solutions exist, the flow proceeds through an ADAF-type flow, as in the classical papers (Abramowicz et al. 1995; Honma 1996; Kato & Nakamura 1998). The second option is based on evaporation of the cold disk caused by electron conduction between the disk and the two-temperature hot corona (Różańska & Czerny 2000b; Meyer & Meyer-Hofmeister 2002).

In the first case the transition from a cold disk to an ADAF flow occurs at

$$\begin{eqnarray}&&{R}_{\mathrm{evap}}=2.0\,{\alpha }_{0.1}^{4}{\dot{m}}^{-2}{R}_{\mathrm{Schw}}\end{eqnarray} \tag{ 3 }$$

(see, e.g., Equation (8) in Czerny et al. 2004), where $\dot{m}$ is the dimensionless accretion rate, α_0.1 is the viscosity parameter in units of 0.1, and R_Schw is the Schwarzschild radius of the black hole (=2${R}_{{\rm{g}}}=2{GM}/{c}^{2}$). Here $\dot{m}$ is defined for a fixed Newtonian efficiency of the accretion process:

$$\begin{eqnarray}&&\dot{m}=\displaystyle \frac{{\dot{M}}_{\bullet }}{{\dot{M}}_{\mathrm{Edd}}};\,\,\,\,{\dot{M}}_{\mathrm{Edd}}=\displaystyle \frac{48\pi {{GM}}_{\bullet }{m}_{{\rm{p}}}}{{\sigma }_{{\rm{T}}}c},\end{eqnarray} \tag{ 4 }$$

where ${m}_{{\rm{p}}}$ is the proton mass and σ_T is the Thomson cross section.

In the description of the second scenario we adopt Equation (11) from Czerny et al. (2004) for the transition radius between the outer disk and an inner hot flow since it contains the effect of the magnetic pressure and compares most favorably with the observed extension of the BLR:

$$\begin{eqnarray}&&{R}_{\mathrm{evap}}=19.5\,{\alpha }_{0.1}^{0.8}{\beta }^{-1.08}{\dot{m}}^{-0.53}{R}_{\mathrm{Schw}},\end{eqnarray} \tag{ 5 }$$

where β is the ratio of the total (gas + radiation) pressure to the total plus magnetic pressure and varies between 0 (magnetic pressure dominance) and 1 (no magnetic pressure). We neglect the emission from the inner hot flow because this very hot plasma, with electron temperature of the order of tens of keV, does not contribute to the 5100 Å flux. This emission can to some extent affect the BLR by Compton-heating the clouds and the intercloud medium, suppressing the line emission. However, since we are interested only in the ionization flux, and not in full radiative transfer computations with cooling/heating balance, we neglect this emission, effectively locating the inner disk radius at R_evap. In this scenario the dependence on the spin practically disappears since the outer disk is only weakly affected by the rotation of the central black hole.

We assume the standard view that the localization of the BLR is related to the ionizing flux. When we use L_ion defined in Equation (1), we follow the approach of Wang et al. (2014a). We assume the scaling of R_BLR with the ionizing flux to have the form of a power law with the same index as determined by Bentz et al. (2013),

$$\begin{eqnarray}&&\mathrm{log}\,{R}_{\mathrm{BLR}}=0.542\mathrm{log}\,{L}_{\mathrm{ion}}+\mathrm{const},\end{eqnarray} \tag{ 6 }$$

where we specifically took the value of the index from their fit Clean. The value of constant has to be adjusted since now we use L_ion instead of L₅₁₀₀ as done in Bentz et al. (2013).

When we use Q as the parameterization of the incident flux, we follow an even more classical approach to modeling of the BLR. It was argued in the past that the BLR properties are well approximated by the fixed value of the ionization parameter, U, and the representative cloud density. The ionization parameter U is defined as

$$\begin{eqnarray}&&U=\displaystyle \frac{Q}{4\pi {r}^{2}{{cn}}_{e}}\end{eqnarray} \tag{ 7 }$$

(see, e.g., Ferland & Netzer 1983), where n_e is the representative local density of the cloud and Q is the number of ionizing photons (above 1 ryd) emitted by the accretion disk. We thus calculate the BLR radius from this formula:

$$\begin{eqnarray}&&\mathrm{log}\,{R}_{\mathrm{BLR}}=0.5\mathrm{log}\,Q+\mathrm{const}.\end{eqnarray} \tag{ 8 }$$

The value of the constant is then related to the universal values of U and n_e. The universal characteristic of the cloud density was argued for at the basis of the radiation pressure confinement mechanism (Baskin & Laor 2018).

We compare the model with the size of the BLR measured as a delay with respect to Hβ line in reverberation campaigns. We use a compilation of the results available in the literature (see Table 1). The measurements come from various groups; most of them were performed for nearby sources. The sample of Bentz et al. (2013) has been carefully corrected for the contamination of the 5100 Å flux by the host galaxy. The sample from Grier et al. (2017) comes from the SDSS–RM (Sloan Digital Sky Survey Reverberation Measurement) project and covers larger redshifts up to z = 1.026. The measurements by Lu et al. (2016) provide an independent determination of the delay in NGC 5548, and Wang et al. (2016) give the delay measured for a gamma-ray-loud NLS1. The sample from Du et al. (2014, 2015, 2016, 2018) represents the SEAMBH (Super-Eddington Accretion in Massive Black Holes) project, so on average these objects have higher Eddington ratios than sources from other samples. In diagrams we mark them with a different color, as they might bias the results. We also give in Table 1 the black hole mass values, taken from the references above, and the Eddington ratio, which we calculate from that mass value and from the monochromatic luminosity assuming a fixed bolometric correction of 9.26 after Shen et al. (2011). Absolute values of the luminosity are given assuming the cosmological parameters: H₀ = 67 km s⁻¹ Mpc⁻¹, Ω_Λ = 0.68, Ω_m = 0.32 (Ade et al. 2014).

Table 1.  Time Delays

Name	log L5100	${R}_{{\rm{H}}\beta }$	$\mathrm{log}\,{M}_{\mathrm{BH}}$	$L/{L}_{\mathrm{Edd}}$	Ref
	($\mathrm{erg}\,{{\rm{s}}}^{-1}$)	(ldt)	M⊙
SDSS J140518	44.33	${41.6}_{-8.3}^{+14.8}$	7.74	0.288	1
SDSS J140759	43.58	${16.3}_{-6.6}^{+13.1}$	7.67	0.059
SDSS J140812	43.15	${10.5}_{-2.2}^{+1.0}$	7.26	0.058
SDSS J140904	44.15	${11.6}_{-4.6}^{+8.6}$	8.45	0.036
SDSS J141004	44.22	${53.5}_{-4.0}^{+4.2}$	8.32	0.058
SDSS J141018	43.58	${16.2}_{-4.5}^{+2.9}$	7.65	0.063
SDSS J141031	44.02	${35.8}_{-10.3}^{+1.1}$	7.91	0.094
SDSS J141041	43.82	${21.9}_{-2.4}^{+4.2}$	7.85	0.070
SDSS J141112	44.12	${20.4}_{-2.0}^{+2.5}$	7.41	0.375
SDSS J141115	44.31	${49.1}_{-2.0}^{+11.1}$	7.94	0.174
SDSS J141123	44.13	${13.0}_{-0.8}^{+1.4}$	7.38	0.411
SDSS J141135	44.04	${17.6}_{-7.4}^{+8.6}$	7.56	0.224
SDSS J141147	44.02	${6.4}_{-1.4}^{+1.5}$	6.95	0.865
SDSS J141214	44.40	${21.4}_{-6.4}^{+4.2}$	7.26	1.019
SDSS J141314	44.52	${43.9}_{-4.3}^{+4.9}$	9.21	0.015
SDSS J141318	43.94	${20.0}_{-3.0}^{+1.1}$	7.51	0.200
SDSS J141324	43.94	${25.5}_{-5.8}^{+10.9}$	8.92	0.008
SDSS J141417	43.40	${15.6}_{-5.1}^{+3.2}$	8.03	0.017
SDSS J141532	44.14	${26.5}_{-8.8}^{+9.9}$	7.23	0.591
SDSS J141606	44.80	${32.0}_{-15.5}^{+11.6}$	9.07	0.040
SDSS J141625	43.96	${15.1}_{-4.6}^{+3.2}$	7.58	0.178
SDSS J141645.15	43.21	${5.0}_{-1.4}^{+1.5}$	7.93	0.014
SDSS J141645.58	43.68	${8.5}_{-1.4}^{+2.5}$	6.90	0.438
SDSS J141706	44.19	${10.4}_{-3.0}^{+6.3}$	6.70	2.258
SDSS J141712	43.21	${12.5}_{-2.6}^{+1.8}$	8.99	0.001
SDSS J141724	43.99	${10.1}_{-2.7}^{+12.5}$	7.57	0.195
SDSS J141729	43.29	${5.5}_{-2.1}^{+5.7}$	8.28	0.008
SDSS J141856	45.38	${15.8}_{-1.9}^{+6.0}$	8.90	0.224
SDSS J141859	44.91	${20.4}_{-7.0}^{+5.6}$	8.05	0.534
SDSS J141923	43.12	${11.8}_{-1.5}^{+0.7}$	7.18	0.065
SDSS J141941	44.52	${30.4}_{-8.3}^{+3.9}$	7.60	0.612
SDSS J141952	44.25	${32.9}_{-5.1}^{+5.6}$	9.23	0.008
SDSS J141955	43.40	${10.7}_{-4.4}^{+5.6}$	7.69	0.037
SDSS J142010	44.09	${12.8}_{-4.5}^{+5.7}$	8.64	0.021
SDSS J142023	44.22	${8.5}_{-3.9}^{+3.2}$	8.58	0.032
SDSS J142038	43.46	${25.2}_{-5.7}^{+4.7}$	7.67	0.045
SDSS J142039	44.14	${20.7}_{-3.0}^{+0.9}$	7.57	0.275
SDSS J142043	43.40	${5.9}_{-0.6}^{+0.4}$	7.20	0.115
SDSS J142049	44.45	${46.0}_{-9.5}^{+9.5}$	9.00	0.020
SDSS J142052	45.06	${11.9}_{-1.0}^{+1.3}$	8.73	0.157
SDSS J142103	43.64	${75.2}_{-3.3}^{+3.2}$	7.89	0.041
SDSS J142112	44.31	${14.2}_{-3.0}^{+3.7}$	8.22	0.092
SDSS J142135	43.47	${3.9}_{-0.9}^{+0.9}$	6.60	0.548
SDSS J142417	44.09	${36.3}_{-5.5}^{+4.5}$	7.70	0.180
Mrk 335	43.76	${14.0}_{-3.4}^{+4.6}$	6.93	0.501	2
Mrk 142	43.59	${6.4}_{-3.4}^{+7.3}$	6.47	0.983
IRAS F12397	44.23	${9.7}_{-1.8}^{+5.5}$	6.79	2.023
Mrk 486	43.69	${23.7}_{-2.7}^{+7.5}$	7.24	0.208
Mrk 382	43.12	${7.5}_{-2.0}^{+2.9}$	6.50	0.312
IRAS 04416	44.47	${13.3}_{-1.4}^{+13.9}$	6.78	3.577
MCG 06	42.67	${24.0}_{-4.8}^{+8.4}$	6.92	0.042
Mrk 493	43.11	${11.6}_{-2.6}^{+1.2}$	6.14	0.694
Mrk 1044	43.10	${10.5}_{-2.7}^{+3.3}$	6.45	0.322
SDSS J074352	45.37	${43.9}_{-4.2}^{+5.2}$	7.93	2.028
SDSS J075051	45.33	${66.6}_{-9.9}^{+18.7}$	7.67	3.359
SDSS J075101	44.18	${30.4}_{-5.8}^{+7.3}$	7.18	0.733
SDSS J075949	44.20	${43.9}_{-19.0}^{+33.1}$	7.44	0.415
SDSS J080101	44.27	${8.3}_{-2.7}^{+9.7}$	6.78	2.257
SDSS J080131	43.97	${11.5}_{-3.7}^{+7.5}$	6.51	2.121
SDSS J081441	43.96	${25.3}_{-7.5}^{+10.4}$	7.18	0.446
SDSS J081456	43.99	${24.3}_{-16.4}^{+7.7}$	7.44	0.259
SDSS J083553	44.44	${12.4}_{-5.4}^{+5.4}$	6.87	2.722
SDSS J084533	44.53	${18.1}_{-4.7}^{+6.0}$	6.76	4.264
SDSS J085946	44.41	${34.8}_{-26.3}^{+9.2}$	7.30	0.952
SDSS J093302	44.31	${19.0}_{-4.3}^{+3.8}$	7.08	1.245
SDSS J093922	44.07	${11.9}_{-6.3}^{+2.1}$	6.53	2.564
SDSS J100402	45.52	${32.2}_{-4.2}^{+43.5}$	7.44	8.802
SDSS J101000	44.76	${27.7}_{-7.6}^{+23.5}$	7.46	1.456
SDSS J102339	44.09	${24.9}_{-3.9}^{+19.8}$	7.16	0.618
NGC 5548	43.21	${7.2}_{-0.3}^{+1.3}$	7.94	0.014	3
1H 0323+342	43.88	${14.8}_{-2.7}^{+3.9}$	7.53	0.164	4
PG 0026+129	44.97	${111.0}_{-28.3}^{+24.1}$	8.15	0.489	5
PG 0052+251	44.81	${89.8}_{-24.1}^{+24.5}$	8.64	0.107
Fairall 9	43.98	${17.4}_{-4.3}^{+3.2}$	8.09	0.058
Mrk 590	43.50	${25.6}_{-5.3}^{+6.5}$	7.55	0.065
3C 120	44.00	${26.2}_{-6.6}^{+8.7}$	7.79	0.122
Ark 120	43.87	${39.5}_{-7.8}^{+8.5}$	8.47	0.018
Mrk 79	43.68	${15.6}_{-4.9}^{+5.1}$	7.84	0.050
PG 0804+761	44.91	${146.9}_{-18.9}^{+18.8}$	8.43	0.223
Mrk 110	43.66	${25.6}_{-7.2}^{+8.9}$	7.10	0.265
PG 0953+414	45.19	${150.1}_{-22.6}^{+21.6}$	8.44	0.408
NGC 3227	42.24	${3.8}_{-0.8}^{+0.8}$	7.09	0.010
NGC 3516	42.79	${11.7}_{-1.5}^{+1.0}$	7.82	0.007
SBS 1116+583A	42.14	${2.3}_{-0.5}^{+0.6}$	6.78	0.017
Arp 151	42.55	${4.0}_{-0.7}^{+0.5}$	6.87	0.035
NGC 3783	42.56	${10.2}_{-2.3}^{+3.3}$	7.45	0.009
Mrk 1310	42.29	${3.7}_{-0.6}^{+0.6}$	6.62	0.035
NGC 4051	41.90	${2.1}_{-0.7}^{+0.9}$	5.72	0.110
NGC 4151	42.09	${6.6}_{-0.8}^{+1.1}$	7.72	0.002
Mrk 202	42.26	${3.0}_{-1.1}^{+1.7}$	6.11	0.104
NGC 4253	42.57	${6.2}_{-1.2}^{+1.6}$	6.49	0.088
PG 1226+023	45.96	${306.8}_{-90.9}^{+68.5}$	8.87	0.918
PG 1229+204	43.70	${37.8}_{-15.3}^{+27.6}$	8.03	0.034
NGC 4593	42.62	${4.0}_{-0.7}^{+0.8}$	7.26	0.017
NGC 4748	42.56	${5.5}_{-2.2}^{+1.6}$	6.61	0.064
PG 1307+085	44.85	${105.6}_{-46.6}^{+36.0}$	8.72	0.098
Mrk 279	43.71	${16.7}_{-3.9}^{+3.9}$	7.97	0.040
PG 1411+442	44.56	${124.3}_{-61.7}^{+61.0}$	8.28	0.141
NGC 5548	43.29	${17.6}_{-4.7}^{+6.4}$	7.94	0.014
PG 1426+015	44.63	${95.0}_{-37.1}^{+29.9}$	8.97	0.033
Mrk 817	43.74	${19.9}_{-6.7}^{+9.9}$	7.99	0.042
Mrk 290	43.17	${8.7}_{-1.0}^{+1.2}$	7.55	0.031
PG 1613+658	44.77	${40.1}_{-15.2}^{+15.0}$	8.81	0.068
PG 1617+175	44.39	${71.5}_{-33.7}^{+29.6}$	8.79	0.029
PG 1700+518	45.59	${251.8}_{-38.8}^{+45.9}$	8.40	1.137
3C 390.3	44.43	${44.5}_{-17.0}^{+27.6}$	9.18	0.013
NGC 6814	42.12	${6.6}_{-0.9}^{+0.9}$	7.16	0.007
Mrk 509	44.19	${79.6}_{-5.4}^{+6.1}$	8.15	0.081
PG 2130+099	44.20	${9.6}_{-1.2}^{+1.2}$	7.05	1.043
NGC 7469	43.51	${10.8}_{-1.3}^{+3.4}$	7.60	0.059
PG 1211+143	44.73	${93.8}_{-42.1}^{+25.6}$	7.87	0.530
PG 0844+349	44.22	${32.3}_{-13.4}^{+13.7}$	7.66	0.265
NGC 5273	41.54	${2.2}_{-1.6}^{+1.2}$	7.14	0.002
KA 1858+4850	43.43	${13.5}_{-2.3}^{+2.0}$	6.94	0.225
Mrk 1511	43.16	${5.7}_{-0.8}^{+0.9}$	7.29	0.055
MCG6-30-15	41.64	${5.7}_{-1.7}^{+1.8}$	6.63	0.008
UGC 06728	41.86	${1.4}_{-0.8}^{+0.7}$	5.87	0.073
MCG+08-11-011	43.33	${15.7}_{-0.5}^{+0.5}$	7.72	0.030
NGC 2617	42.67	${4.3}_{-1.4}^{+1.1}$	7.74	0.006
3C 382	43.84	${40.5}_{-3.7}^{+8.0}$	8.67	0.011
Mrk 374	43.77	${14.8}_{-3.3}^{+5.8}$	7.86	0.061

References. (1) Grier et al. 2017; (2) Du et al. 2014, 2015, 2016, 2018; (3) Lu et al. 2016; (4) Wang et al. 2016; (5) Bentz et al. 2013.

Download table as:  ASCIITypeset images: 1 2 3

Since our sample is larger than the sample considered in Wang et al. (2014a), we first use the same method as was used there, with BLR distance measured from ionizing flux L_ion estimated at 912 Å (see Equation (1)), and without any evaporation effect. The results are shown in Figure 2.

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We see that the change of the sample essentially affects the conclusion reached by Wang et al. (2014a). Their conclusion was that all the monitored objects must have predominantly high spin since at that time most of the measured delays were located along the $\tau \propto {L}_{5100}^{1/2}$ line. Currently, with the presence of many shorter lags in the sample, good coverage of the measured distribution of the time delays is achieved if we allow a whole range of black hole spins from 0 to maximally rotating black holes. A few objects require counterrotating spin. What is interesting, however, is that these objects have rather moderate luminosities ($\sim {10}^{44}$ erg s⁻¹), relatively low Eddington ratios (∼0.05), and black hole masses that are also moderate (∼10⁸–10⁹ M_⊙). These values are roughly consistent with the parameters of the sources in the SDSS–RM sample of Grier et al. (2017). Such accretion rates are relatively low for a quasar sample, but this is the consequence of the selection effect: the monitoring was short, about a year, so the delays were measured only for the sources with delays shorter than 100 days in the observed frame, so only for the low-luminosity tail of the sample. The model predicts even shorter time delays than measured for more massive sources, but observationally determined delays do not populate this region. This is because we allowed for even lower Eddington ratios (0.01) and larger masses (10¹⁰ M_⊙) in our parameter grid than present in the sample.

Next, we use the R_BLR predictions based on the number of photons Q intercepted by the BLR (see Equation (2)), again without any evaporation effect, and the results are shown in Figure 3. The model predictions are qualitatively similar, but not identical. Overall, the Q prescription gives a much stronger bending effect for a counterrotating accretion disk than the model based on L_ion for the same parameters. The trend reverses for a high-spin corotating accretion disk, when a smaller departure from a power-law trend is seen for the Q-based model. This is related to the relativistic effects. For large black hole spin in a corotating disk the radiation emitted at higher inclination angles is strongly beamed and enhanced in comparison with the continuum measured by an observer, which compensates for the spectral bending seen in Figure 1. This also means that Q is not strictly proportional to L_ion. We illustrate this effect in Figure 4. The departure from the strict linearity between the two quantities results both from the derivation of Q as an integral and from the relativistic effects (a difference between photons going to the observer and photons going toward the BLR). The approach based on Q is more accurate, and the examples calculated later on are based on this assumption, but overall the difference between L_ion and Q predictions is not large, and the L_ion approach is equally useful for statistical analysis of the samples.

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Now we compare Q-based delay predictions to the measured time delays (see Figure 3). This approach does not require counterrotating black holes to explain even the shortest time lags. Therefore, with very simple and basic assumptions (standard Novikov–Thorne disk, corotating with a black hole, with a range of masses, accretion rates, and spins, BLR responding to the ionizing continuum at fixed ionization parameter and density), we are able to reproduce fully satisfactorily the observed distribution of the time delays.

However, the assumption that the cold Keplerian disk extends all the way down to the ISCO is under discussion, particularly for lower Eddington ratio sources. If the inner hot flow develops, this part of the disk is no longer a source of UV photons but instead, filled with a hot plasma at electron temperatures of order of 100 keV, is a source of X-ray emission. Effectively this decreases the number of photons available for ionizing hydrogen. Available models allow us to determine the position of this transition and thus are the subject of tests if the predictions are consistent with the observed time delays.

We first test the prediction of the model based on the strong ADAF principle described by Equation (3). For the viscosity parameter, we assume the value α = 0.02, which was suggested by direct studies of the quasar UV variability (Siemiginowska & Czerny 1989; Starling et al. 2004) and is consistent with damped random walk results (Kelly et al. 2009; see also the discussion in Grzȩdzielski et al. 2017). The results are shown in Figure 5. In this case no counterrotating spin values are necessary, and the whole plane is well covered even if all the objects have high spin (left panel). The longest delays for a given value of L₅₁₀₀ require high spin values, while shorter delays may imply either lower spin or lower Eddington ratio. This degeneracy can be removed if we actually have reliable determination of the Eddington ratio for an individual object. However, in comparison with the predictions for the disk without an inner cutoff, the requested accretion rates of sources with short delays are much higher, and all observed sources are then predicted to have Eddington ratios above ∼0.05. Some of the Eddington ratios in the SDSS–RM sample may be lower than this limit if the bolometric luminosity is estimated as roughly 9 times the L₅₁₀₀ luminosity and the black hole mass measurement from Grier et al. (2017) is adopted. On the other hand, such an estimate of the bolometric luminosity is highly uncertain since it does not take into account the differences in the spectral shapes clearly seen in Figure 1.

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Finally, we test the solutions based on the evaporation of the inner disk as described by Equation (5). The results obtained for the parameters $\alpha =0.02$ and β = 0.5 (moderate magnetic field strength; see Equation (5)) are far from being satisfactory (see Figure 6). The longest time delays are not reproduced, even if we allow for high spin since the transition radius between the outer cold disk and an inner hot flow is far too large. A significant contribution of the magnetic field to the total pressure in the disk is inconsistent with the observational data. We thus decreased the role of the magnetic field, assuming β = 0.99 (i.e., only 1% of the contribution from the magnetic pressure to the gas plus radiation pressure), but such a parameter adjustment still did not fully solve the problem (see Figure 7). We thus additionally allowed for a significant decrease of the viscosity parameter down to the value α = 0.001, and this allowed to reconcile the model and the data (see Figure 8). For these parameters, evaporation of the outer cold disk is indeed inefficient. The transition radius between the outer cold disk and an inner hot flow takes place at 5.7R_Schw for the Eddington ratio 0.01, and it is closer in or absent for larger accretion rates and/or lower spin. When we assumed the expected values of the viscosity parameter α = 0.02, the transition radii start at 62.3R_Schw (for the lowest Eddington ratio 0.01). This low viscosity α ∼ 0.001 is thus strongly required if the measured time delays are to be consistent with the disk evaporation phenomenon.

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However, as we mentioned before, such small values of the viscosity parameter are not supported by the quasar variability (see Grzȩdzielski et al. 2017). This implies that the physically based description of the disk evaporation and the transition to inner hot flow still requires a more advanced approach. The issue is difficult and has been the subject of studies for several years in the context of cataclysmic variables (Meyer & Meyer-Hofmeister 1994), binary black holes, and AGNs (Liu et al. 1999; Różańska & Czerny 2000a). The transition process is further complicated by the possibility of the existence of the inner cool disk separated by the gap of pure hot flow from the outer cold disk (Liu et al. 2006; Meyer et al. 2007; see Meyer-Hofmeister & Meyer 2014; Meyer-Hofmeister et al. 2017; Taam et al. 2018 for recent developments). Here we considered only the inner radius of the outer cold disk.

Emission lines of the BLR respond to the emission originating close to the black hole, as clearly seen from the response of the line with respect to the variable incident flux. Traditionally, this response was modeled by parameterizing the incident continuum with a single parameter in the form of the ionization parameter, or ionizing flux. Then, if a range of radii and densities were allowed, like, for example, in the LOC model (Baldwin et al. 1995), line ratios were successfully modeled. However, this general approach did not give a direct insight into the reason for the presence of the numerous clouds above the disk. Wind models are also parametric and not quite constraining the cloud formation mechanisms. The observational discovery of the scaling of the BLR size with a monochromatic flux instead of the ionizing flux opened a way for testing the cloud origin. Such a scaling, with a very small dispersion, is consistent with the dust-based formation mechanism of the BLR (Czerny & Hryniewicz 2011; Czerny et al. 2015, 2017; Baskin & Laor 2018), at least with respect to the Hβ formation region. In this model the BLR position is controlled not by the ionizing flux but by the availability of the material for the irradiation, lifted above the disk by the dust-driven radiation pressure. This scaling also strongly disfavored self-gravitational instability as the BLR origin (Czerny et al. 2016).

New observational data presented in Figure 2 drastically change this view. Now the dispersion around the delay–monochromatic flux relation is very large, and most of the new measurements lie below the previous tight scaling law of Bentz et al. (2013). Part of the outliers come from the super-Eddington ratio sample by Du et al. (2014, 2015, 2016, 2018), and in this case the departure from the previous scaling can be explained as a departure from the thin Keplerian disk approximation for the incident continuum. The average value of the Eddington ratio in these sources is 1.72. However, other numerous points come from the SDSS–RM sample of Grier et al. (2017), and these objects have low to moderate Eddington ratios, with the average value of 0.24, not significantly higher than in the Bentz et al. (2013) sample (0.14), well within the dispersion in the two samples.

The measurements we use here come from the literature, as described in Section 2.4, and were not prepared in a uniform way. All sources in the Bentz et al. (2013) sample were carefully corrected for the host galaxy contamination using Hubble Space Telescope (HST) observations, and the corrections were important not just for the faintest nearby AGN but also for PG quasars (Bentz et al. 2006). A good example from Bentz et al. (2013) is the source SBS 116+583A (z = 0.02787), where the AGN emission contributes only 12% to the total flux at 5100 Å, i.e., not correcting this source for starlight gives a shift of 0.92 in the logarithm of L₅₁₀₀, or, equivalently, a factor of 2.9 in the expected time delay. The sources from Du et al. (2014) were also corrected for the starlight using HST images, while for the sources reported later (Du et al. 2015, 2016, 2018) the authors used an empirical relation from Shen et al. (2011). Grier et al. (2017) used the information on host contamination from Shen et al. (2015), who employed principal component analysis and the measurements of the measured stellar velocity dispersion to decompose the spectra into AGNs and host components. The accuracy of such methods is difficult to estimate. The error in the host subtraction can be responsible for the incorrect determination of L₅₁₀₀.

New results, if reliable, take us all the way back to the original concept of the simple BLR response to the ionizing flux. The range of delays is well consistent with the predictions of the ionized continuum based on the thin Keplerian disk since this ionized continuum does not scale linearly with the monochromatic flux at 5100 Å owing to the curvature of the spectrum in the UV band (see Figure 1). Models based on the dusty origin of the BLR seem not justified, as in this case no departure from the Bentz et al. (2013) scaling is predicted for a broad range of black hole masses and accretion rates. Instead, the observed delays are consistent with a universal ionization parameter and universal density in the Hβ formation region for a very broad range of parameters.

The range of spin plays an important role, and all spins from nonrotating black holes to maximally spinning ones (a = 0.998) are requested to create the appropriate representation of the reverberation studied sources. Counterrotating disks are not strongly required, but a small fraction of such sources is not excluded. Evaporation of the inner disk and a transition to ADAF are not really required for Eddington ratios above 1% studied here, and models that predict inner ADAF flow at such high Eddington ratios are disfavored, so the models with an inner cold disk separated from the other cold disk sound more attractive. However, the predictions for such models are more complex and were not tested in detail in the current paper. In this paper we also did not test again the self-gravity scenario, but since self-gravity in general predicts shorter delays than the Bentz et al. (2013) relation, it remains to check whether this option offers equally good coverage of the parameter space as a simple universal ionization parameter with the universal density model.

Recently, a new model of the BLR origin was suggested by Wang et al. (2017). In this model, clumps in the dusty/molecular torus are tidally captured and disrupted by the central black hole. A population of inflowing clouds forms at the first stage, a tiny fraction of clumps is channeled into outflows (less than 10%), and then most of the disrupted clumps form a BLR disk with Keplerian rotation (virialized component). Unlike the dust-based clouds (Czerny & Hryniewicz 2011), the supply of the BLR clouds originates from the dusty torus. The virialized component in this model can produce the canonical R–L relation for sub-Eddington AGNs provided that the ionization parameter, density, and temperature are universal. Additionally, the infalling component is supported by PG 2130+099 (see Section 4.3). This model avoids difficulties of the disk self-gravity model or dust-based failed winds.

What we found in this paper is the role of black hole spin in the observed R–L relation jointly with accretion rates. The evidence in support of retrograde accretion onto black holes has very important implications for cosmological evolution of black holes. If the black holes are fueled in a stochastic manner, with no preferred orientation, they are slowly spinning owing to cancellation of random angular momentum of accreted gas (King et al. 2008). Wang et al. (2009) built up an equation of the radiative efficiency (η) across cosmic time from observed data of galaxy and quasar surveys. The authors derived the evolutionary curve and obtained that the radiative efficiency changes from η ≈ 0.3 at redshift z ≈ 2, where quasar density peaks, down to 0.03 at low redshift. This supports the role of episodic accretion at later stages of the galaxy evolution. The downsizing behavior of spins was further discussed by Li et al. (2012). Subsequently, Volonteri et al. (2013) found a similar behavior of spin evolution from numerical simulations. The sensitive dependence of the R–L relation on spin offers a new tool of estimating black hole rotation, in particular for those AGNs with the retrograde accretion disks, which may have too weak gravitational effects on iron Kα line profiles to measure their spins from X-ray observations.

To draw firm conclusions, however, the measured delays have to be accurate. The SDSS–RM campaign concentrated on average on higher-redshift sources, the campaign was relatively short, and the observational cadence was not very dense. Determination of the time delays in AGNs is not very straightforward since the variability has a red-noise character and the reprocessing region is extended. Frequently, two peaks show up in the cross-correlation function (e.g., Du et al. 2016), and if the campaign is too short, only one solution (the shorter one) may be found, although it may not be actually the correct one. Therefore, an extension of this campaign is clearly necessary to ensure that the measured delays are not affected by the way in which they are performed.

The cadence selected for the RM campaign is very important. A low-candence campaign will smear short-timescale variations, so that the measured lags tend to be longer. PG 2130+099 is an example. Kaspi et al. (2000) measured an Hβ line lag of 188 days with a low cadence of about ∼20 days and a couple of seasonal gaps, with the total campaign duration as long as about 8 yr. With a cadence of a few days, however, Grier et al. (2008) measured a lag of ∼23 days with large uncertainties (but the total length in their case was only about 100 days). Moreover, Grier et al. (2012) got a lag of ∼10 days with a cadence of 1 day. Hu et al. (2018) (ApJ, submitted) measured a lag of ∼24 days with a cadence of 3 days, confirming the results of Grier et al. (2008), but found that the lag of ∼188 days follows the dust reverberation scaling relation (Koshida et al. 2014), suggesting that the reverberations with a lag of 188 days are from the inner edge of the torus. This supports the ideas of Wang et al. (2017). Additionally, PG 2130+099 is a super-Eddington source, and the other two lags of ∼10 and ∼24 days can be explained by the self-shadowing effects of slim accretion disks (Wang et al. 2014b). Overall, the BLR is extended, and the choice of the cadence can focus the monitoring on a particular part of the BLR, making a comparison of results for different sources very difficult. Also, the nonlinear response of the BLR to the line emission, particularly if the characteristic variability timescale is short in comparison to the average time delay, can easily lead to apparent shortening of the time delay, as pointed out by Goad & Korista (2014). Future RM campaigns should be planned very carefully with respect to the cadence.