ACCRETION DISK TEMPERATURES OF QSOs: CONSTRAINTS FROM THE EMISSION LINES

The idea of accretion disks around supermassive black holes as the power source of active galactic nuclei (AGN) is widely accepted. Physical plausibility, the lack of successful alternatives, and the frequent presence of bipolar jets support this picture. Qualitatively, the blue-ultraviolet "Big Blue Bump" in the spectral energy distribution (SED) of QSOs resembles the expected thermal emission from the disk atmosphere (Shields 1978; Malkan 1983). Models achieve some success in fitting the SED of individual QSOs (e.g., Hubeny et al. 2001). However, the simple disk model has known shortcomings in accounting for the soft X-ray continuum, optical polarization, variability (e.g., Koratkar & Blaes 1999). Further difficulties include (1) the "ionization problem" involving a deficit in the number of hard ionizing photons in the extrapolation of observed UV continuum slopes (e.g., Netzer 1985; Korista et al. 1997), (2) the "size problem" resulting from microlensing measurements giving optical continuum sizes larger that expected from disk theory (Pooley et al. 2007; Morgan et al. 2010; Jiménez-Vicente et al. 2012), and (3) the "shape problem" in which the observed ultraviolet continuum slope is similar among AGN and softer than expected. In particular, observed SEDs typically show a break to steeper slopes around 1100 Å rest wavelength (Shang et al. 2005 and references therein). For further discussion of these issues, see Lawrence (2012 and references therein).

These results have raised serious questions about the detailed applicability of standard disk models. Motivated by the availability of black hole masses in AGN estimated from broad line widths, Bonning et al. (2007, hereinafter B07) examined the optical and ultraviolet colors of QSOs in the Sloan Digital Sky Survey (SDSS)⁵ in the context of disk theory. We defined a characteristic temperature (here denoted T_ref) expected on the basis of M_BH and the accretion rate $\dot{M}$ (see the detailed discussion below). Comparing T_ref with quasar continuum colors, B07 found poor agreement between observed color trends and those predicted by standard disk models (a shift toward bluer colors at higher temperatures). For luminosities approaching the Eddington limit, the observed colors actually become redder with increasing T_ref, in qualitative disagreement with expectations. B07 suggested that the discrepancy might result from a modification of the inner disk at radii where radiation pressure strongly affects the disk vertical structure. Modified disk models with no emission inside a critical radius gave improved agreement with observation.

In an effort to gain additional perspective on the correspondence between disk theory and observation, we here extend the results of B07 to include the QSO emission lines as diagnostics of the higher frequency portion of the disk continuum. The emission lines are governed by the intensity of the continuum at ionizing frequencies. The emitted flux at these frequencies is more sensitive to the accretion disk temperature and potentially offer a stronger test of the trend of the SED of QSOs with T_ref. We again use the large dataset of quasar spectra available from the SDSS. We compare the observed quasar spectra to model predictions computed with the code AGNSPEC (Hubeny et al. 2001) coupled with the photoionization code Cloudy (Ferland et al. 1998). With a large range of T_ref, one might expect strong trends in emission-line ratios and equivalent widths as a function of disk temperature. We find that these expectations are not borne out in observed QSO spectra.

For the sake of economy, we refer the reader to B07 for further background and references to earlier work. We assume a concordance cosmology with Ω_Λ = 0.7, Ω_m = 0.3, and H₀ = 70 km s⁻¹ Mpc⁻¹.

2.1. Accretion Disk Properties

In standard thin-disk theory (Shakura & Sunyaev 1973), accreting matter gradually spirals inwards as viscous stresses transport angular momentum outward. These stresses lead to local dissipation of energy that flows vertically to the surface of the disk and is radiated by the photosphere. The resulting effective temperature, given by F = σ T_eff⁴, shows a radial dependence T_eff∝r^−3/4 at radii substantially larger than the inner boundary of the disk. As the inner boundary is approached, the effective temperature reaches a maximum T_max and then drops to zero at the inner boundary if the inner boundary condition is one of zero torque. For a disk radiating locally as a blackbody, the disk spectrum is fixed by T_max, for a given value of black hole spin. For realistic opacities, there are significant spectral features such as Lyman edges of hydrogen and helium, either in emission or absorption (Hubeny et al. 2001) that can cause disks with the same value of T_max to differ in their detailed continuum spectra. However, T_max still remains a useful sequencing parameter.

The key parameters for a QSO accretion disk are the black hole mass and spin, the accretion rate, and the observer viewing angle. An important reference luminosity is the Eddington limit,

$$\begin{equation} L_\mathrm{Ed} = \frac{4\pi cG{M_{\rm BH}} }{\kappa _{e}}=(10^{46.10}\, {\rm erg\,s^{-1}})M_8, \end{equation} \tag{ 1 }$$

where κ_e is the electron scattering opacity per unit mass and M₈ = M_BH/(10⁸ M_☉). The total luminosity produced is

$$\begin{equation} L_{\rm bol} = \epsilon \dot{M} c^{2} = (10^{45.76}\, {\rm erg\,s^{-1}})\epsilon _{-1} {\dot{M}_0}. \end{equation} \tag{ 2 }$$

Here $\dot{M}_{\mathrm{0}}$ is the accretion rate in M_☉ yr⁻¹, and = 10₋₁ increases from 0.057 for a non-rotating hole to 0.31 for a rapidly rotating Kerr hole with angular momentum parameter a_* = 0.998 (Novikov & Thorne 1973; Thorne 1974). For a_* = 0.998

$$\begin{equation} {T_{\rm max}} =(10^{5.56}\, {\rm K})M_{8}^{-1/4}(L_{\mathrm{bol}} /L_{\mathrm{Ed}})^{1/4}, \end{equation} \tag{ 3 }$$

or alternatively, ${T_{\rm max}} =(10^{5.54}\,{\rm K})M_{8}^{-1/2}\, L_{46}^{1/4}$, where L₄₆ ≡ L_bol/(10⁴⁶ erg s⁻¹). For a Schwarzschild hole, T_max is cooler by a factor of 10^0.46 for a given M_BH and L_bol (e.g., Shields 1989). We also define a reference accretion rate ${\dot{M}_{\mathrm{Ed}}} \equiv L_{\mathrm{Ed}} /c^2 = (10^{-0.66}\, M_\odot \,\mathrm{yr}^{-1})M_8$ and a dimensionless accretion rate ${\dot{m}} \equiv {\dot{M}} /{\dot{M}_{\mathrm{Ed}}}$. Note our definition of $\dot{M}_{\mathrm{Ed}}$ in terms of = 1, so that a value ${\dot{m}} = 3.1$ gives L_Ed for a_* = 0.998.

2.2. Deriving M_BH and T_ref

Following B07, we use a sequencing parameter T_ref equal to the value of T_max given by Equation (3) for a given M_BH and $\dot{M}$ (assuming a_* = 0.998). We emphasize that the difficulties described above already suggest that actual disk temperatures depart from theoretical expectations, and that T_ref serves here as a sequencing parameter by which to look for qualitative trends that may or may not conform to the predictions of standard disk theory. Derivation of M_BH from the width of the Hβ or Mg ii broad emission lines has become an accepted approximation in recent years. The FWHM of the broad lines is taken to give the circular velocity of the broad-line emitting material (with some geometrical correction factor). The radius of the broad-line region (BLR), determined from echo mapping studies, increases as a function of the continuum luminosity, R∝L^Γ. Bentz et al. (2009) find Γ = 0.52 ± 0.07, consistent with the value 0.5 suggested by photoionization physics. Here we use (Shields et al. 2003)

$$\begin{equation} {M_{\rm BH}} = (10^{7.69}\,M_\odot)v_{3000}^2 L_{44}^{0.5}, \end{equation} \tag{ 4 }$$

where v₃₀₀₀ ≡ FWHM/3000 km s⁻¹, and L₄₄ ≡ λL_λ(5100)/(10⁴⁴ erg s⁻¹).

The derived value of T_ref depends almost entirely on the broad line FWHM for a given object. The bolometric luminosity can be estimated as L_bol = f_L × λL_λ(5100), following Kaspi et al. (2000). Using Equation (4) and this expression for L_bol, Equation (3) becomes

$$\begin{equation} {T_{\rm ref}} = (10^{5.43}\,\mathrm{K}) v_{3000}^{-1} L_{44}^{-(\Gamma - 0.5)/2} (f_L/9)^{-1/4}. \end{equation} \tag{ 5 }$$

We use a bolometric correction factor f_L = 9 (Kaspi et al. 2000; Richards et al. 2006).

2.3. SDSS Spectra and T_ref Bins

We use the SDSS DR5 spectroscopic data base for "quasars" as our primary observational material. We selected AGN using an automated spectrum fitting program and imposing quality cuts as described by Salviander et al. (2007). These fits give values for quantities such as the flux and width of the broad Hβ line and the continuum flux at various rest wavelengths including λ5100. The adopted redshift range is z = 0.1–0.8 so as to keep the [O iii] nebular lines out of the telluric water vapor region. Approximately one-half of the objects fall above the QSO/Seyfert galaxy luminosity boundary at log νL_ν(5100) = 44.44 in units of erg s⁻¹. Following B07, we derive M_BH and T_ref for the individual quasars as described above, and then group them into bins with log T_ref ranging from 5.0 to 5.7 defined by windows of 4.95 to 5.05, etc. Table 1 gives the average values of log M_BH and log νL_ν(5100) for these bins. Also given is the accretion rate and resulting log T_ref on the assumption f_L = 9 and a_* = 0.998, based on the bin averages for log M_BH and log νL_ν(5100). (An alternative approach to the accretion rate is discussed below.) The values of log M_BH systematically decrease from 8.27 to 7.20 as log T_ref increases from 5.0 to 5.7. The mean luminosity and accretion rate differs only slightly among the bins. As a result, the Eddington ratio increases with log T_ref from log L/L_Ed = −1.5 to −0.3 across the bins.

Table 1. Properties

Tref	MBH	νLν(5100)	${\dot{M}}$	L/LEd	${\dot{M}} _{5100}$	(L/LEd)5100
5.0	8.72	44.39	−0.95	−1.47	−0.77	−1.34
5.1	8.51	44.36	−0.98	−1.30	−0.63	−0.99
5.2	8.31	44.33	−0.97	−1.13	−0.49	−0.65
5.3	8.12	44.34	−0.95	−0.93	−0.29	−0.26
5.4	7.90	44.30	−1.00	−0.75	−0.12	0.13
5.5	7.66	44.22	−1.06	−0.59	−0.03	0.46
5.6	7.43	44.13	−1.13	−0.44	0.06	0.78
5.7	7.20	44.04	−1.19	−0.30	0.15	1.10

Notes. Mean properties of the SDSS quasars in the various T_ref bins. All quantities are log₁₀, with M_BH in solar masses and $\dot{M}$ in solar masses per year. The subscript "5100" refers to AGNSPEC models with accretion rate adjusted to match observed continuum at 5100 Å. See the text for discussion.

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For each of the bins, we form a composite spectrum giving equal weight to all objects based on the frequency-integrated flux at earth over the spectrum. These composite spectra were measured to give values of several observational quantities (e.g., the [Ne v] emission-line intensity) used in the analysis below. Line measurements were carried out in IRAF.⁶

Note that the high values of T_ref given by Equation (5) for the SDSS quasars already conflict with observed SEDs. As noted above, SEDs typically turn over around 1100 Å. Observed EUV energy distributions fall far below the predictions of blackbody or disk models for such high temperatures, as illustrated in Figure 1 of Lawrence (2012).

2.4. Accretion Disk Models

Here we discuss tests of the success of the standard thin accretion disk model to explain a variety of observed properties of AGN. We discuss two different approaches for determining the accretion rate of the disk, which, together with M_BH, determines T_ref. First, we follow B07 in assuming a universal bolometric correction f_L = 9 as described above. Then we discuss an alternative approach in which $\dot{M}$ is adjusted in the disk models so as to reproduce the observed value of νL_ν(5100) (cf. Figure 1).

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3.1. Emission-line Intensities

As found above, typical values of T_max for the accretion disk (for the parameters of our sample) are of the order of 200,000 K, corresponding to a blackbody peak at hν ≈ 3kT ≈ 1.5hν_H, where hν_H = I_H is the hydrogen ionization potential. Thus, the output of the disk in hydrogen ionizing photons, Q(H⁰), will be sensitive to the value of T_ref, and the sensitivity will be greater for ions with higher ionization potentials. The amount of ionized gas and the resulting line luminosities, as well as the relative intensities of lines from different levels of ionization, should depend strongly on T_ref. If the actual disk temperature is less than T_ref, the sensitivity of the ionizing output to any differential changes in that temperature as a function of T_ref will be even greater.

3.1.1. Hβ Equivalent Width

A basic application of this concept is the equivalent width of the broad Hβ line. For a photoionized nebula, the luminosity of the Hβ line is given by radiative recombination theory (Osterbrock & Ferland 2006) as

$$\begin{equation} L(\rm {H}\beta) = (\Omega /4\pi) (\epsilon _{\rm {H}\beta }/\alpha _{\mathrm{B}}) Q({\rm H}^0). \end{equation} \tag{ 6 }$$

Here, Ω/4π is the covering fraction for the BLR, _Hβ is the recombination-line emission coefficient, α_B is the radiative recombination coefficient for hydrogen to quantum levels n = 2 and higher, and Q(H⁰) is the ionizing photon luminosity (photons per second). The Hβ equivalent width is then given by EW = L(Hβ)/L_λ(4861 Å). As T_max increases, we expect a strong increase in EW(Hβ), because the ionizing frequencies are more sensitive to disk temperature than the optical continuum emission. Figure 2 shows the predicted EW for the set of AGNSPEC models described above, on the assumption of a covering fraction of 0.5 independent of T_max. The expected increase of EW with T_max is evident. (The covering fraction of 0.5 was chosen to give agreement with the observed EW of Hβ at log T_ref = 5.0 and is reasonable in the context of the unified model of AGN.)

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Figure 2 also shows the observed trend of the Hβ EW as a function of T_ref for our SDSS composites. The observed values show a progressive decrease from 120 Å to 50 Å as log T_ref increases from 5.0 to 5.7. This trend is in qualitative disagreement with the model prediction. While it is possible that the covering fraction changes systematically with T_ref, an order-of-magnitude decrease in Ω/4π would be required from log T_ref = 5.0 to 5.7. A search for independent signatures of such a trend may be worthwhile. However, it seems likely that such an extreme trend in covering fraction would have been noticed in prior investigations. Therefore, the discrepancy shown in Figure 2 reinforces the case that something is wrong with the standard disk model in regard to the ionizing photon output. This discrepancy resembles that found in B07 in which the continuum colors do not show the expected trend toward bluer colors at higher T_ref. The systematic decrease in Hβ equivalent width with increasing T_ref gives a new context to the long known correlation between the EW and FWHM of the broad Hβ emission line (Boroson & Green 1992), since the derived T_ref depends mainly on the Hβ broad line width (see Equation (5)). While there were already reasons to doubt that AGN disks achieve temperatures as high as T_ref, it is striking that the observed equivalent width actually trends in a direction contrary to the expected trend with T_ref.

Equation (6) does not allow for collisional excitation or other processes that can affect Hβ in the BLR. In order to estimate the magnitude of these effects, we computed a series of models for the BLR using version 10.0 of the Cloudy photoionization code, most recently described by Ferland et al. (1998). The input ionizing spectrum was the output of the AGNSPEC program as described above. Following Ferland et al. (2009), the BLR was modeled as an optically thick slab with hydrogen density N_H = 10¹⁰ cm⁻³ and incident ionizing flux ϕ = 10¹⁹ cm⁻² s⁻¹, giving an ionization parameter U = ϕ/Nc = 10^−1.5. Equation (6) corresponds to a predicted flux F(Hβ) = (_Hβ/α_B)ϕ. The model results exceeded this prediction by 0.08 and 0.17 dex for log T_max = 5.0, respectively. These values and their trend with T_max do not alter the basic conclusions reached above.

3.1.2. Ionization Ratios

Typical AGN spectra show emission lines from ions with a large range of ionization potentials. In the case of ionized nebulae, a useful diagnostic of the ionizing continuum is the ratio of the lines of He ii and H i. In a normal nebular ionization structure, He ii occupies an inner volume of the Strömgren sphere (or a surface layer of a slab) whose fraction of the total H⁺ volume is proportional to the ionizing photon luminosity ratio, Q(He⁺)/Q(H⁰). Here Q(He⁺) is the ionizing photon luminosity above 4 Rydbergs frequency. The recombination line ratio $I({\rm He\,{\scriptsize II}}\,\lambda 4686)/I(\rm {H}\beta)$ is in turn proportional to the nebular average of N(He⁺²)/N(H⁺), giving

$$\begin{equation} I(\lambda 4686)/I(\rm {H}\beta) = 1.02\, Q({\mathrm{He}}^+)/Q({\rm H}^0). \end{equation} \tag{ 7 }$$

This assumes that the nebula is optically thick to the Lyman continuum.

Figure 3 shows that Q(He⁺)/Q(H⁺) does indeed increase strongly with increasing T_max in the AGNSPEC models. Figure 4 shows the composite spectra for the various bins in log T_ref. The broad He ii line is blended with Fe ii and difficult to measure. The He ii line in the narrow line spectrum is well defined for our cooler composites, in which the broad line widths are large. However, it is difficult to separate the narrow from the broad components of λ4686 for the hotter composites, which have relatively narrow broad lines. Nevertheless, two points can be made. A subjective inspection of the figure indicates little trend in the narrow He ii strength as a function of log T_ref. Also, the observed values of narrow He ii/Hβ for the lower T_ref bins are larger than would be expected for the predicted disk continuum, absent an additional hard component such as a power-law; and there is little trend with increasing T_ref.

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For an observationally more feasible test, we consider the narrow line ratio of [Ne v] λ3426 to [Ne iii] λ3869. The Ne⁺⁴ ion requires even higher ionizing photon energies than He⁺² and thus the [Ne v]/[Ne iii] ratio should be even more sensitive to T_max than He ii/Hβ. For illustration, we computed a series of Cloudy models for the Narrow Line Region (NLR). The input ionizing spectrum was the AGNSPEC continuum for T_max corresponding to the various adopted values of T_ref. The NLR was modeled as an optically thick slab with hydrogen density N_H = 10^4.0 cm⁻³ and incident ionizing flux ϕ = 10^12.5 cm⁻² s⁻¹, giving an ionization parameter U = ϕ/Nc = 10^−2.0. The default chemical abundances were used, including log N(A)/N(H) = −1.00, −3.31, −4.0 for He, O, and Ne, respectively. Figure 5 shows the predicted [Ne v]/[Ne iii] ratio as a function of T_max. The models show the expected strong trend with T_max, with [Ne v] being very weak for the cooler models.

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However, the observed values of [Ne v]/[Ne iii] show no significant trend with T_ref. Thus, this observed line ratio, involving emission lines from the NLR rather than the BLR, also fails to support the expectation of a harder spectrum in the ionizing ultraviolet for higher T_ref.

3.2. Soft X-Rays

AGN spectra characteristically include emission in the hard and soft X-ray bands. In general, this emission involves frequencies too high to be the result of ordinary thermal continuum emission from a standard accretion disk. Laor et al. (1997) and others have noted the inconsistency of observed soft X-ray fluxes with the simple thin-disk model. A Comptonizing corona, producing a power-law spectrum extending to high photon energies, is sometimes invoked (Czerny & Elvis 1987). However, for the higher values of T_ref that we consider here, substantial soft X-ray emission from the inner disk is predicted, and there is the possibility of overproducing soft X-rays. We have computed T_ref in the manner described above, for the AGN in the sample described by Laor et al. (1997). Laor gives X-ray fluxes and values of the optical (3000 Å) to soft X-ray (0.3 keV) spectral index defined by $L_{\mathrm{s}}/L_{\mathrm{o}} = (\nu _s/\nu _o)^{\alpha _{{\rm os}} }$. Figure 6 shows the predicted value of α_os as a function of T_max for a simple set of AGNSPEC models with M_BH = 10⁸ M_☉. The increase in α_os with increasing T_ref is evident. In contrast, the observed values of α_os in the Laor sample, grouped into our same bins in T_ref, show no significant trend with T_ref. In particular, for log T_ref > 5.2, the models predict a greater value of L_s/L_o than observed. This is significant, because one might invoke a Comptonized component to explain the soft X-rays for low T_max; but the hotter disks, in the standard model, overpredict the soft X-rays even before any Comptonized component is added. While the origin of the X-ray flux of AGN remains uncertain, it is remarkable that the observed values of α_os show no trend with T_ref. Thus, we have in α_os yet another qualitative discrepancy between observed AGN properties and the predictions of the simple disk model.

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The dependence of α_os on disk effective temperature has been previously discussed by other authors. For example, Deroches et al. (2009) in their Figure 6 show α_os versus log[(L_bol/L_Ed)^1/4M_BH^1/4]. Aside from minor differences in the derivation of L_bol and M_BH, their temperature axis can be calibrated to our log T_ref scale by adding 7.56. The SDSS objects in their plot span a similar range in T_max to ours, and likewise show little systematic trend in α_os with T_ref.

3.3. Models with Variable Bolometric Correction

B07 found that observed continuum colors of SDSS QSOs did not become progressively bluer with increasing T_ref in the manner predicted by the AGNSPEC models. They suggested that this resulted from higher Eddington ratios at higher T_ref, leading to departures from the standard thin disk model. This might involve thickening by radiation pressure, inefficient "slim disk" accretion, and mass loss in a radiation-driven wind (see also Deroches et al. 2009). As an estimate of the potential effect on the continuum colors, we computed AGNSPEC models in which the inner disk was truncated inside the radius where radiation pressure gave a vertical scale height H for the disk, relative to the radius, of H/R = 0.5. This led to improved agreement with the observed colors, in particular giving a non-monotonic trend of color with T_max.

Here we explore a more physical model in which the disk effective temperature is modified inside the critical radius. The basic idea, following Poutanen et al. (2007) and Begelman et al. (2006), is to take into account the luminosity produced at a given radius relative to L_Ed. If the luminosity emitted by a wide annulus, d ln L/d ln R, is well below L_Ed, then the disk will not be severely thickened by radiation pressure, and standard disk physics applies. However, except near the inner boundary, d ln L/d ln R increases with decreasing R. For sufficiently large values of ${\dot{M}} /{M_{\rm BH}}$, a radius is reached where d ln L/d ln R reaches L_Ed. Near this radius, serious modifications of the disk structure can be anticipated. Shakura & Sunyaev (1973) called this the "spherization radius" R_sp on the basis that radiation pressure would severely thicken the disk. Poutanen et al. (2007) find that, inside R_sp, a combination of wind mass loss and advection occurs to a degree that keeps the local luminosity d ln L/d ln R close to L_Ed. This results in a break in the run of effective temperature with radius, so that inside R_sp one has T_eff∝R^−1/2 rather than the standard dependence T_eff∝R^−3/4. This reduces the value of T_max actually reached in the inner disk as well as the luminosity radiated from the hottest parts of the disk.

The value of R_sp in units of the gravitational radius is approximately $r_\mathrm{sp} \equiv R_\mathrm{sp} /R_\mathrm{g} \approx {\dot{m}}.$ As $\dot{M}$ increases, the effective temperature at R_sp actually decreases, following

$$\begin{equation} T_\mathrm{sp} = (10^{5.75}\,{\rm K}){\dot{m}} ^{-1/2}M_8^{-1/4}. \end{equation} \tag{ 8 }$$

Here we have used the Newtonian expression for the local flux, omitting the normal factor of three increase in flux due to viscous transport of energy from smaller radii. This modification was found by Poutanen et al. (2007) because of energy loss in the inner disk due to advection and mass loss. For large values of $\dot{m}$, a luminosity ∼L_Ed is emitted in the vicinity of R_sp, and a similar luminosity is emitted per unit lnR at smaller radii. Thus T_sp substantially controls the character of the emitted spectrum. Hence, the fact that T_sp decreases with increasing $\dot{m}$ once spherization comes into play is suggestive of the reversal in the trend of colors with T_ref found in B07. (Note that, in this context, T_max is defined in terms of the accretion rate entering the disk at large radii, and differs from the actual maximum temperature in the inner disk.)

The potential for an inverse relationship between $\dot{M}$ and disk color temperature is strengthened if one considers the optical depth of the disk wind. In the case of highly super-Eddington accretion, Poutanen et al. (2007) find that a large part of the inflowing material is expelled as a wind in the vicinity of R_sp. In the simple approximation of a spherical wind with constant radial velocity equal to the circular orbital velocity at R_sp, the optical depth of the wind, measured radially from R_sp, is $\tau _\mathrm{sp} \approx {\dot{m}} ^{1/2}$. The optical depth in the wind, measured outward from some radius R that exceeds R_sp, is τ(> R) ≈ τ_sp(R/R_sp)⁻¹. For the radius of the photosphere, where τ(> R) = 1, this gives $r_\mathrm{ph} \approx {\dot{m}} ^{3/2}$. The effective temperature at the photosphere is

$$\begin{equation} T_\mathrm{ph} = (10^{5.75}\,{\rm K}){\dot{m}} ^{-3/4}m_8^{-1/4}. \end{equation} \tag{ 9 }$$

This raises the prospect that, for super-Eddington accretion rates, the maximum visible disk temperature will be T_ph rather than T_max. An estimate of the progression of the disk temperature as it actually appears to the observer is then

$$\begin{equation} T_\mathrm{app} = \mathrm{min}({T_{\rm max}}, T_\mathrm{ph}). \end{equation} \tag{ 10 }$$

This assumes that there is enough absorption opacity in the wind to reprocess the emerging energy; otherwise, the escaping spectrum may still be characterized by T_sp.

For our Series 1 models above, with an accretion rate based on f_L = 9, the value of log L/L_Ed reaches −0.3 for log T_max = 5.7. Thus, unless opacity sources other than electron scattering reduce the effective Eddington limit, the effects of radiation pressure may be modest. However, for our Series 2 models with $\dot{M}$ adjusted to fit the continuum at λ5100, log L/L_Ed reaches +1.1 for log T_max = 5.7. For this series, at log T_max  =  (5.0, 5.4, 5.7) we have $\mathrm{log}\, {\dot{m}} = (-0.79, +0.29, +1.65)$; log min(T_max, T_sp) = (5.05, 5.44, 5.13), and log min(T_max, T_ph) = (5.05, 5.37, 4.75). Thus, a reversal of the trend of the apparent temperature of the disk as a function of the nominal T_max would be expected at log T_max ≈ 5.4. The fact that this value agrees with the turn-around in the color trend found in B07 is fortuitous, given the approximations in our discussion. However, this simple calculation does illustrate the possibility of a reverse trend of color temperature with ${\dot{M}} /{M_{\rm BH}} ^2$ for super-Eddington rates.

In order to explore the impact of super-Eddington accretion on the disk spectrum and resulting emission-line intensities, we have computed modified models with AGNSPEC in which the standard T_eff inside a break radius R_b is multiplied by a factor (R/R_b)^1/4, following the discussion above. For the value of the break radius, we took R_b/r_g = x_b  (L/L_Ed). We present results for x_b = 7, which was motivated by a preliminary analysis but which serves here simply as an illustrative example. We used the same values of M_BH and $\dot{M}$ as used in our Series 1 and Series 2 models above. The run of T_eff with radius for the modified models is shown in Figure 7. As expected, there is little effect for Series 1 (f_L = 9); but substantial alterations are seen for Series 2. In Figure 2, the Series 2 prediction for the Hβ equivalent width levels off above log T_max = 5.4. This reduces but does not eliminate the discrepancy with the observed trend. Figure 3 shows a similar leveling of the predicted He ii intensity. Figure 5 shows a substantial reduction in the predicted Ne v intensity for log T_max around 5.4. These models are not intended to achieve a detailed fit to the observed spectra, but they do illustrate the substantial effect on the ionizing continuum that results from a modified disk structure resulting from super-Eddington accretion.

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Our focus has been on the entire SDSS quasar data set, which is strongly dominated by radio-quiet objects. However, radio loudness raises interesting questions regarding the black hole spin, which affects the inner disk boundary and thus the maximum temperature reached in the disk. Numerous authors have discussed the possibility that radio-loud AGN have rapidly spinning black holes (e.g., Blandford 1990), and on the other hand Garofalo et al. (2010) have proposed that radio-loud objects have retrograde black hole spin with respect to the disk angular momentum. The innermost stable circular orbit (ISCO) decreases from 6R_g for a = 0 to 1.22R_g for a_* = 0.998. Correspondingly, the radius of maximum effective temperature moves inward from 9.5R_g to 1.55R_g; and the value of T_max increases from $10^{4.79}{\dot{M}_0} ^{1/4} M_8^{-1/4}$ to $10^{5.43 }{\dot{M}_0} ^{1/4} M_8^{-1/4}$ (Shields 1989). For retrograde spin, the ISCO moves still farther out. These differences in the inner disk temperature could have a significant effect on the ionizing spectrum of the disk, while having less effect on the optical luminosity. Thus, we might expect differences between radio-loud and -quiet AGN in the observational quantities sensitive to T_max discussed above.

We have formed composite spectra separately for the radio-loud and radio-quiet AGN, adhering to our previously defined bins in T_ref. Here radio-loud was taken simply to mean detected in the Faint Images of the Radio Sky at Twenty Centimeters (FIRST) radio survey (Becker et al. 1995). This is not equivalent to the standard definition (Kellermann et al. 1989) and is distance-dependent, but it is sufficient to look for any qualitative difference between radio-loud and -quiet AGN. Figure 8 shows the observed trend of the [Ne v] intensity with T_ref for the radio-loud and -quiet composites. The [Ne v]/[Ne iii] ratio is similar for the two groups for log T_ref = 5.0, but there is a decline for the radio-detected objects with increasing T_ref. Certainly, there is no evidence that [Ne v] is stronger for the radio-detected objects, as might be expected if they have higher spin of a prograde sense. On the other hand, the weaker [Ne v] of the radio objects might be expected if they involve retrograde disks.

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The results presented here give focus to concerns about the disk model for the AGN continuum that have been voiced in the literature by numerous authors (see Section 1). In general terms, AGN over a wide range of M_BH and luminosity have rather similar SEDs (see Laor & Davis 2011 and references therein). Organization of the observational data in terms of T_ref helps to clarify what trends are expected on the basis of standard disk theory. We find that the expected trends in the intensity of Hβ, He ii, and [Ne v] are not seen in the observations. Over a range of a factor of five in the value of T_ref, the predicted disk temperature based on M_BH and νL_ν(5100), the [Ne v] intensity shows no systematic variation, and the Hβ equivalent width shows a reverse trend. While our models are highly simplified, in particular for cases that approach or exceed the Eddington limit, the basic immunity of these key line ratios as well as α_os to the value of T_ref, which is essentially 1/FWHM(Hβ), is a striking empirical fact. These results reinforce earlier studies indicating that a serious re-examination of the structure and emission mechanisms of the AGN central engine is in order. This might reasonably start with more detailed studies of the nature of disks in systems with high Eddington ratios, taking account of advection, winds, and all contributing sources of opacity.

We thank the anonymous referee for a careful and thorough review that greatly improved the final manuscript. We thank Ivan Hubeny for the use of the AGNSPEC program, R. Antonucci, O. Blaes, A. Laor, and B. Wills for helpful discussions. Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.