APPLICATION OF THE DISK EVAPORATION MODEL TO ACTIVE GALACTIC NUCLEI

We investigate stationary accretion to a non-rotating black hole. The accretion takes place via a standard optically thick and geometrically thin disk (Shakura & Sunyaev 1973) enclosed by a hot, accreting corona. Such a corona may form by processes similar to those operating in the surface of the Sun, or from a thermal instability in the uppermost layers of the disk (e.g., Shaviv & Wehrse 1986). Both the disk and corona are individually powered by the release of gravitational energy associated with the accretion of matter affected through viscous stresses. The interaction between the corona and the underlying disk results from the vertical conduction of heat by electrons, mass evaporation/condensation and hence enthalpy flow, and inverse Compton scattering of the disk photons by coronal electrons. The disk evaporation/condensation model takes account of both the mass exchange (mass evaporation/condensation) and energy exchange (conduction and enthalpy flux) between the disk and corona. This is in contrast to classical disk corona models where only radiative coupling is considered (e.g., Haardt & Maraschi 1991). Assuming a radial one-zone approximation, we investigate the mass flow and energy balance in the corona and calculate the detailed vertical structure and radiation emitted from the corona and the disk as functions of the viscosity parameter in the corona, black hole mass, and mass accretion rate.

For the modeling of the accretion process, the interaction between the disk and corona is an important and distinctive process compared to the case for an ADAF or a disk alone. The corona is an ADAF-like accretion flow modified by the vertical heat conduction and inverse Compton scattering of disk photons, which play a key role in cooling the electrons as a consequence of the soft photon flux from the underlying disk. The ions in the corona are directly heated by viscous dissipation, partially transferring their energy to the electrons by means of Coulomb collisions. The energy gained by electrons cannot be effectively radiated away and is conducted to the lower, cooler, and denser coronal layers by electron–electron collisions. In the transition layer, the conductive heat flux is radiated away through bremsstrahlung only if the number density in this layer reaches a critical value. If the density is too low to efficiently radiate the energy (lower than the critical value), a certain amount of lower, cooler gas is heated up, whereby the bremsstrahlung cooling rate is raised, until an energy equilibrium is established between the conduction, radiation, and heating of the cool gas. The transfer of gas from the disk to the corona, to establish the equilibrium, is called evaporation. It occurs in the low-density corona. If the density is so high that bremsstrahlung is more efficient than the conductive heating, a certain amount of gas is over-cooled and condenses onto the disk until energy equilibrium is reached. This condensation process takes place in the high-density corona. Efficient cooling processes such as strong inverse Compton scattering of the disk photons can also facilitate such condensation.

The gas evaporating into the corona retains angular momentum and differentially rotates around the central object. By frictional stresses, the gas loses angular momentum and drifts inward in such a way that the corona continuously drains gas toward the central object. The coronal accretion flow is resupplied by continuous evaporation and, therefore, steady flows are established in the disk and the corona with the mass exchange between the two flows.

In the disk corona model, we assume that the accreting gas is cold and thus enters the disk rather than the corona as occurs in a binary system where the mass transferred from the donor star is constrained to the orbital plane. That is, we do not consider the possibility that the accreting gas enters the disk as a corona (see discussion in Section 4.2), which may be appropriate for the Galactic center or for very low-luminosity AGNs. Hence, the model is restricted to systems in the low/hard state and not necessarily to the very low quiescent state. Within our approximation, the coronal accretion is supplied by evaporating disk matter at distances remote from the accreting black hole. The mass flow rate through the corona at a given distance is the sum of the mass evaporation from the outer edge of the disk inward to a given distance. Hence the mass flow rate in the corona increases toward the black hole as a result of the radial contribution to the mass evaporation, notwithstanding the possible loss of gas due to disk outflows. Note that it is possible that the mass flow rate contributed by evaporation in the outer region is sufficiently high that evaporation ceases at some radius and condensation takes place in the inner region.

In the following, the equations that include the detailed physical processes described above are given. The model used here is based on Liu et al. (2002a) and includes modifications (Meyer-Hofmeister & Meyer 2003) and updates incorporated in recent years (e.g., Qian et al. 2007; Qiao & Liu 2009).

Equation of state:

$$\begin{equation} P={\Re \rho \over 2\mu } (T_{i}+T_{e}) \end{equation} \tag{ 1 }$$

Equation of continuity:

$$\begin{equation} \centering {d\over dz}(\rho v_z)=\eta _M{2\over R}\rho v_R -{2z\over R^2+z^2}\rho v_z \end{equation} \tag{ 2 }$$

Equation of the z-component of momentum:

$$\begin{equation} \rho v_z {dv_z\over dz}=-{dP\over dz}-\rho {GMz\over (R^2+z^2)^{3/2}} \end{equation} \tag{ 3 }$$

The energy equation of the ions

$$\begin{eqnarray} &&\hspace{-1.5pc}{d\over dz}\left\lbrace \rho _i v_z \left[{v^2\over 2}+{\gamma \over \gamma -1}{P_i\over \rho _i}-{GM\over (R^2+z^2)^{1\over 2}}\right]\right\rbrace \nonumber\\ &&\hspace{-0.8pc}={3\over 2}\alpha P\Omega -q_{ie}\nonumber\\ &&\hspace{-0.1pc}+\,{\eta _E}{2\over R}\rho _i v_R \left[{v^2\over 2}+{\gamma \over \gamma -1}{P_i\over \rho _i}-{GM\over (R^2+z^2)^{1\over 2}}\right]\nonumber\\ &&\hspace{-0.1pc}-\,{2z\over {R^2+z^2}}\left\lbrace \rho _i v_z \left[{v^2\over 2}+{\gamma \over \gamma -1}{P_i\over \rho _i}-{GM\over (R^2+z^2)^{1\over 2}}\right]\right\rbrace, \end{eqnarray} \tag{ 4 }$$

where q_ie is the energy exchange rate between the electrons and the ions,

$$\begin{equation} {q_{ie}}={\bigg ({2\over \pi }\bigg)}^{1\over 2}{3\over 2}{m_e\over m_p}{\ln (\Lambda) }{\sigma _T c n_e n_i}(\kappa T_i-\kappa T_e) {{1+{T_*}^{1\over 2}}\over {{T_*}^{3\over 2}}}, \end{equation} \tag{ 5 }$$

with

$$\begin{equation} T_*={{\kappa T_e}\over {m_e c^2}}\bigg (1+{m_e\over m_p}{T_i\over T_e}\bigg). \end{equation} \tag{ 6 }$$

The energy equation for both the ions and the electrons is

$$\begin{eqnarray} &&\hspace{-1.6pc}{\frac{d}{dz}\left\lbrace \rho {v}_z\left[{v^2\over 2}+{\gamma \over \gamma -1}{P\over \rho } -{GM\over \left(R^2+z^2\right)^{1/2}}\right] + F_c \right\rbrace }\nonumber\\ &&\hspace{-0.8pc}=\frac{3}{2}\alpha P{\Omega }-n_{e}n_{i}L(T_e)-q_{\rm Comp}\nonumber\\ &&\hspace{-0.1pc}+\,\eta _E{2\over R}\rho v_R \left[{v^2\over 2}+{\gamma \over \gamma -1}{P\over \rho } -{GM\over \left(R^2+z^2\right]^{1/2}}\right]\nonumber\\ &&\hspace{-0.1pc}-\,{2z\over R^2+z^2}\left\lbrace \rho v_z\left[{v^2\over 2}+{\gamma \over \gamma -1}{P\over \rho }- {GM\over \left(R^2+z^2\right)^{1/2}}\right] +F_c\right\rbrace,\nonumber\\ \end{eqnarray} \tag{ 7 }$$

where n_en_iL(T_e) is the bremsstrahlung cooling rate, of which the radiative cooling function L(T_e) is taken from Raymond et al. (1976). Here, q_Comp is the Compton cooling rate

$$\begin{equation} {q_{\rm Comp}}={4\kappa T_e\over m_e c^2}n_e\sigma _T c u, \end{equation} \tag{ 8 }$$

where u is the energy density of the soft photon field. The thermal conduction flux, F_c, is given by (Spitzer 1962)

$$\begin{equation} F_c=-\kappa _0T_e^{5/2}{dT_e\over dz}, \end{equation} \tag{ 9 }$$

with κ₀ = 10⁻⁶erg s^-1cm⁻¹K^−7/2 for a fully ionized plasma.

All parameters in the above equations are in cgs units and are defined as follows. Specifically, P, ρ, T_i, and T_e are the pressure, density, ion temperature, and electron temperature, respectively. The vertical and radial speed of the mass flow are denoted by v_z and v_R. In Equation (1), μ = 0.62 is the mean molecular weight assuming a standard chemical composition (X = 0.75, Y = 0.25) for the corona. The number density of the ions, n_i, is assumed equal to that of the electrons, n_e, for convenience, which is strictly true only for a pure hydrogen plasma. In the continuity and energy equations, the terms starting with 2z/(R² + z²) derive from the gradual expansion of the vertical flow channel with height as the geometry changes from cylindrical to spherical, and η_M is the mass advection modification term, and η_E is the energy modification term. These terms parameterize the difference between lateral inflows and outflows. We take η_M = 1 for the case without consideration of the effect of mass inflow from the outer neighboring zone of the corona, and η_E = η_M + 0.5 is a modification to the previous energy equations (for details, see Meyer-Hofmeister & Meyer 2003). The other quantities are as follows: G is the gravitational constant, M is the mass of the accreting black hole, m_p and m_e are, respectively, the mass of the proton and the electron, κ is the Boltzmann constant, c is the speed of light, a is the radiation constant, σ is the Stefan-Boltzmann constant, σ_T is the Thomson-scattering cross section, γ = 5/3 is the ratio of specific heats, and ln(Λ) = 20 is the Coulomb logarithm.

The five differential equations, Equations (2), (3), (4), (7), and (9), which contain five variables P(z), T_i(z), T_e(z), F_c(z), and $\dot{m} (z)(\equiv \rho (z) v_z)$, can be solved with five boundary conditions. In particular, at the lower boundary of the interface between the disk and corona, the conductive flux is exactly radiated away, and there is no downward heat flux. The temperature of the gas should be the effective temperature of the accretion disk. Previous investigations (Liu et al. 1995) show that the coronal temperature increases from the effective temperature to 10^6.5 K in a very thin layer and that the conductive flux can be expressed as a function of pressure at this temperature. Thus, the lower boundary conditions can be reasonably approximated (Meyer et al. 2000a) as

$$\begin{equation} T_i=T_e=10^{6.5}\;{\rm K},\ {\rm and} \ F_c=-2.73\times 10^6 P\ {\rm at}\ z=z_0. \end{equation} \tag{ 10 }$$

At infinity, there is no artificial confinement and hence no pressure. This requires a sonic transition at some height z = z₁. As there is no heat flux from/to infinity, we constrain the upper boundary as

$$\begin{equation} F_c=0\ {\rm and} \ v_z^2=V_s^2\equiv P/\rho ={\Re \over 2\mu }(T_i+T_e)\ {\rm at}\ z=z_1. \end{equation} \tag{ 11 }$$

With the above boundary conditions, we assume a set of trial lower boundary values for P and $\dot{m}$ to start the integration along z. Only when the trial values for P and $\dot{m}$ fulfill the upper boundary conditions can the presumed P and $\dot{m}$ be taken as true solutions of the differential equations.

In order to reveal the basic properties of disk corona in AGNs, we make some simplifications and approximations. Specifically, we assume an α prescription for the viscosity. In reality, the disk corona structure depends on the magnitude and functional form of the viscosity. This is likely due to magnetic effects associated with the magnetorotational instability (see Balbus & Hawley 1991), though its effectiveness in providing a sufficiently large α remains to be understood (see King et al. 2007). The magnetic field also influences the rate of heating (via non-ideal effects associated with magnetic reconnection) and cooling of the system since non-thermal radiation processes associated with synchrotron radiation and synchrotron self-Compton emission could be operating. In addition, the field can reduce the effectiveness of the vertical heat conduction. Thus, the field likely plays an important role in determining the structure of the disk corona and emission features if it is sufficiently strong. The irradiation from the hot corona could also become important in heating the disk if the coronal radiation is strong compared to the disk. This may occur at accretion rates when the disk starts to be truncated. Its affect is relatively unimportant at very low accretion rates when the disk can be truncated and most of the coronal radiation can escape. In addition, at very high accretion rates, the corona irradiation is not as important as the disk radiation itself, although there are indications that the coronal emission can be comparable to the optical emission in some AGNs. The irradiation can also be important if most of the accretion energy is released in the corona (e.g., Haardt & Maraschi 1991; Nakamura & Osaki 1993; Kawaguchi et al. 2001), but the nature of the mechanism resulting in such a hypothesis remains unknown. Therefore, we do not consider the effect of irradiation from the inner corona in our level of approximation. With these qualifications, we study the vertical structure in detail, determining the radial flow owing to the vertically distinct flows in the context of the lower cool disk and upper hot coronal flows.

As described above, the evaporation process diverts the accretion flow from the disk to the corona. Consequently, for accretion rates within the disk lower than the maximal evaporation rate, the disk is completely evaporated within a specific region. This region is filled with coronal gas (see Figure 2), and the accretion takes place in a geometrically thick flow. On the other hand, for accretion rates higher than the maximal evaporation rate, the optically thick disk cannot be significantly depleted at any distance. Thus, the disk extends to the ISCO with an overlying accreting coronal flow coexisting with the disk, fed by continuous evaporation. The variation of the accretion flow geometry with the accretion rate is illustrated in the right panel of Figure 2.

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Such an evaporation feature has been successfully applied to interpret the spectral states observed in BHXRBs, which are believed to be caused by the different accretion modes. That is, for luminosities greater than the critical luminosity, the accretion is dominated by the mass flow in the disk with the radiative spectrum described by a multi-color blackbody. The system is in the so-called soft state. In contrast, for luminosities lower than the critical value, the accretion is dominated by the corona/ADAF, resulting in a hard state spectrum. Therefore, the evaporation model predicts a spectral transition from an ADAF to a disk-dominated accretion flow at a critical luminosity corresponding to the maximal evaporation rate. The prediction for AGNs is similar to BHXRBs despite the vast difference in the masses of the black holes because the evaporation feature is independent of the black hole mass.

The value of the maximal evaporation rate is known to be dependent on the accretion history. In a low/hard state, the radiation from the truncated disk is much less than that from the ADAF/corona, and Compton cooling in the corona due to the disk photons can be neglected. Thus, the transition from a low/hard state to a high/soft state can be appropriately described by the solid curve in the left panel of Figure 3. That is, the transition takes place at a luminosity of 0.027L_Edd assuming a standard viscosity, α = 0.3. However, the transition from the soft to hard states takes place at a relatively low accretion rate in comparison to the transition from the hard to soft states. This is because the inverse Compton scattering of the disk photons from the ISCO region during the soft state serves as an additional cooling agent for the coronal gas, which leads to a diminished evaporation rate. In this case, Compton cooling must be included in the energy Equation (7). The radiation flux, F_r = cu, from the central region as seen by the corona at a distance R is given by

$$\begin{equation} F_r={L\over 4\pi R^2}{2z\over (R^2+z^2)^{1/2}}, \end{equation} \tag{ 12 }$$

where L is the luminosity from the central source, which is related to the central accretion rate by assuming an energy conversion efficiency of 0.1, $L=0.1\dot{M} c^2$.

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Equation (12) reveals that the coronal structure and evaporation rate now depend on the accretion rate. Based on calculations, it was found that for accretion rates near the values corresponding to the hard-to-soft transition rate, the maximal evaporation rate is less than the accretion rate. Hence, the disk cannot be truncated at this accretion rate level. However, the evaporation rate increases with decreasing accretion rates (i.e., decreasing soft photon flux), and once the accretion rate decreases to $\dot{m}\approx 0.005$, the maximal evaporation rate equals the accretion rate. This signifies the onset of disk truncation by evaporation at $\dot{m}\approx 0.005$ during the decay of the outburst, as shown by the dashed line in the left panel of Figure 3. For even lower accretion rates, the disk is truncated, and the accretion flow is dominated by the ADAF/corona. Therefore, the soft-to-hard state transition takes place at $\dot{m} \approx 0.005$, which is ∼5 times lower than the accretion rate for the hard-to-soft state transition. This provides a natural explanation for the so-called luminosity hysteresis observed in BHXRBs (Meyer-Hofmeister et al. 2005; Liu et al. 2005). In AGNs, the different transition luminosities in the soft-to-hard and hard-to-soft states imply that objects with luminosities in the range 0.005 < L/L_Edd < 0.027 can be in either the soft spectral state or the hard spectral state. Specifically, if objects are in the rising phase, their spectra are hard; if they are in the decaying phase, their spectra are soft.

This range in luminosity provides a framework by which objects showing a soft spectrum (big blue bump and steep X-ray spectrum) can be observed at luminosity levels that are comparable to objects showing a hard spectrum (without a big blue bump and with a flat X-ray spectrum). Since the accretion timescale is long compared to the observation timescale, the limited observations do not distinguish whether an object is in the rising or decaying phase. The model provides a potential method to determine the phase of variation if the luminosity level lies between the transition luminosities. That is, for an object with 0.005 < L/L_Edd < 0.027 exhibiting a hard spectrum, it is either a persistent source or, if transient, in the rising phase, and a transition to a soft state is expected to follow, while a soft spectrum indicates that the object is in the decaying phase, with a subsequent transition to a hard state expected. The overlap in the Eddington ratio for the soft and hard spectral states predicted by our model could also lead to a misclassification of AGN subtypes.

To summarize, the evaporation model predicts that objects characterized by luminosities less than ∼0.005L_Edd are in the low/hard states, whereas those objects with luminosities greater than ∼0.027L_Edd are in the high/soft state. Within the luminosity range 0.005L_Edd and 0.027L_Edd, the objects can be either in the high/soft state or in the low/hard state, depending on their secular variability. Objects in this luminosity range may undergo variation with possible transition to another state.

Observationally, spectral state transitions are difficult to detect, although extensive studies on nearby galaxies (for a review, see Ho 2008, 2009) suggest that the galactic nuclei evolve through distinct states in response to changes in the mass accretion rate. In the sample of Ho (2009), objects are largely systems in the low or quiescent state, for which the Eddington ratios and corresponding accretion rates are all below the critical accretion rate predicted by the disk evaporation model.

Finally, we point out that the quantitative predictions of the evaporation model depend on the viscosity parameter, α, chosen to be 0.3. Larger values of α result in higher evaporation rates and smaller transition radii. Here, the greater viscous heating leads to greater conduction of heat to the disk surface, facilitating evaporation. As a result, the critical luminosities of the soft-to-hard and the hard-to-soft transition also depend on α. For example, the critical luminosities of the hard-to-soft transition range from 1% to 10% of the Eddington luminosity for 0.2 < α < 0.6 (Qiao & Liu 2009).

Observational estimates for α in binary star systems have been summarized in King et al. (2007), showing viscosity parameters to be 0.2 ≲ α ≲ 0.6 (values are converted according to the definition ν = 2/3αc_sH as used here). However, such estimates for the AGN are severely lacking. In a few cases, constraints have been established. For example, the investigation of optical variability in AGNs (Starling et al. 2004) suggests a very low viscosity parameter, 0.01 ≲ α ≲ 0.03, if the optical emission is generated in a standard, fully ionized thin disk, and the variability timescale is attributed to the disk thermal timescale. Nevertheless, these values are noted to be lower limits because shorter timescales may have been missed (Starling et al. 2004).

Broad lines are a distinguishing feature of type 1 AGNs. Hints concerning their origin and nature have been inferred from the empirical relation between the size of broad-line region (BLR) and its bolometric luminosity, R_BLR ∝ L^1/2_bol, commonly believed to be a consequence of photoionization of the BLR gas by continuum radiation emitted from the accreting central regions. The formation of the BLR has been connected with radiatively driven winds launched from the disk surface (Murray & Chiang 1997). If the BLR is, indeed, associated with the disk through winds, the truncation of the disk leads to the truncation of the BLR.

An essential feature of the disk evaporation model is the truncation of a disk for accretion rates below a critical value. For decreasing rates below this value, the inner radius of the disk increases. This property predicts the existence of a region where broad lines would be excluded in a plane described by the line width and Eddington ratio. The variation of the truncation radius with respect to the Eddington ratio from the numerical calculations is illustrated in Figure 4, where it is assumed that $L_{\rm bol}/L_{\rm Edd}=\dot{m}$ for simplicity. If a lower efficiency of energy conversion is applied to lower accretion rates, the curve is flatter at lower Eddington ratios. The model predicts that the BLR lies to the above/right of the truncation curve. The region lying below the curve corresponds to the region of exclusion.

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Three truncation curves are plotted in Figure 4, showing the dependence of the region of exclusion on the viscosity and evolutionary stage of an AGN. The solid (α = 0.3) and dotted (α = 0.2) lines delineate this region for objects during the rise in outburst. The difference between these two curves indicates the sensitivity for the location of the excluded region boundary to the viscosity parameter. The dash-dotted curve represents the case for objects during decay from a soft state for a standard viscosity parameter α = 0.3, which reveals the dependence of the exclusion region boundary on the evolutionary history. Therefore, the model predicts a region of exclusion for the occurrence of broad lines under the dash-dotted curve if a standard viscous parameter is adopted, irrespective of the temporal variation of the AGN. In the regime between the solid and dash-dotted curve, broad lines can be present only for objects decaying from a soft state.

To compare with observations, we take the full sample of 35 reverberation-mapped AGNs (Kaspi et al. 2000, 2005; Peterson et al. 2004) after optical correction of the host-galaxy star light (Bentz et al. 2006, 2009). The size of the BLR, in units of Schwarzschild radius, versus the Eddington ratio is plotted in Figure 4. The empirical relations of the BLR size with respect to the Eddington ratio are also plotted in the figure for different black hole masses. It can be seen that most of the data fall into the region between the empirical lines for M = 10⁷ M_☉ and M = 10⁹ M_☉ since this is the main range of black hole masses of the sample. However, at low luminosities (L_bol/L_Edd ≲ 0.01), the BLR size does not continually decrease along the empirical lines. Therefore, we suggest that the BLR, which is described by the empirical relation, is "truncated" at low Eddington ratios to a distance corresponding to the inner edge of the evaporation-truncated disk. With decreasing Eddington ratios, the disk recedes outward and hence the inner edge of the BLR increases until the ionization of the supposed BLR is insufficient for the emission of broad lines. The data in this small sample are in approximate agreement with this prediction, though additional data at low luminosities are necessary to confirm it.

A detailed investigation of the connection between the BLR and disk truncation was also carried out in an earlier work by Czerny et al (2004) for a very large sample of AGNs. It was found that the strong ADAF principle, based on the hypothesis that accretion is via an ADAF whenever an ADAF can exist, better describes the observational data than the disk evaporation model. We point out that a very small value of the viscosity parameter (0.02 for evaporation model and 0.04 for ADAF model) was required in that study, which is not necessary in the present study. Upon comparison of the two versions of evaporation model, we find that our detailed numerical calculations of the vertical structure of the disk corona exhibit a dependence of the truncation radius on the viscosity and magnetic fields, which differs from the approximation-based generalized model (Ròżańska & Czerny 2000b). In particular, a large deviation occurs at the Eddington ratio close to the critical transition value. Specifically, our model reveals that a small change in α results in a large change in the critical Eddington ratio characterizing disk truncation ($\dot{m}_{\rm crit} \propto \alpha ^{2.34}$), as illustrated in Figure 4 for α varying from 0.2 to 0.3. This sensitivity allows for the possibility of an inner disk and hence very broad lines at low Eddington ratios by decreasing α only slightly. In contrast, the weak dependence of the truncation radius on α as used in Czerny et al. (2004) leads to a very small value for α in order that broad lines be present at low Eddington ratios. With our detailed numerical model, we expect that the observational sample adopted by Czerny et al. (2004) can also be explained provided that α ∼ 0.1. This implies that it is unnecessary to assume a very small α when the detailed disk evaporation model is used to explain the broad-line emissions. In this sense, the disk evaporation model is not as restrictive as compared to models based on the strong ADAF principle.

We note that the theoretical curves are nearly parallel at low Eddington ratios and that the truncation radii are not as sensitive to α in this regime as compared to Eddington ratios of ∼0.01. For a normal viscosity value and assuming an appropriate ionization, Figure 4 shows that the BLR can be present at Eddington ratios as low as ∼0.001. However, the size of the BLR is not as small as described by the empirical law since the evaporation truncates the inner part or all of the BLR at low Eddington ratios. Therefore, the disk evaporation model predicts two regimes in which the broad lines are "narrower:" one corresponding to very low L/L_Edd where the disk is truncated at large distances associated with a large BLR (i.e., LLAGNs; for a review, see Ho 2008); the other is at very high L/L_Edd where a full disk is present, but its partially ionized region is at large distances (such as the NLS1s). Caution is necessary because there exist uncertainties in the observational data and the adopted parameters in spectral fitting, both of which could lead to large error bars in the derived accretion rate and truncation radius.

For very low accretion rates, the disk is truncated at large distances by evaporation. For instance, the disk is truncated at 6600R_S at $\dot{m}=0.001$ (in the case of standard viscosity α = 0.3 and no magnetic fields). If regions at such large distances can still emit broad lines, the emission line would be quite narrow. The FWHM of emission lines, as estimated by the Keplerian velocity with a correction factor $2/\sqrt{3}$ (which accounts for velocities in three dimensions and the full width corresponding to twice the velocity dispersion), is given by

$$\begin{equation} {\rm FWHM}={2\over \sqrt{3}}\sqrt{GM\over R_{\rm BLR}}=\sqrt{2c^2\over 3R_{\rm BLR}/R_S}. \end{equation} \tag{ 13 }$$

Thus, at R_BLR = 6600R_S, the FWHM of emission line is ∼3000 km s⁻¹. If we take 2000 km s⁻¹ as the critical line width between broad-line AGNs and narrow-line Seyfert 1 galaxies, we obtain an inner radius of the BLR as 15, 000R_s. To truncate the disk to this distance, the accretion rate should be 4 × 10⁻⁴ in units of the Eddington rate as derived from the disk evaporation model. At even lower accretion rates, the disk is truncated at a larger radius and, hence, the BLR would no longer be recognizable. Therefore, the disk corona evaporation model predicts that the lower limit to the luminosity for the inference of a BLR is ∼0.0004–0.001L_Edd.

The lower limit to the luminosity of AGNs characterized by a BLR implies that the orientation-based unification model is not applicable to LLAGNs. Instead, objects with Eddington ratios lower than ∼0.001 may intrinsically lack a BLR rather than simply be obscured by a torus. In other words, the model predicts that there exist so-called "true Seyfert 2" galaxies. This is supported by the observations of Bianchi et al. (2008), who discovered an unobscured Seyfert 2 galaxy (Sy2), NGC 3147, without a BLR. The Eddington ratio for this object ranges roughly between 8 × 10⁻⁵ and 2 × 10⁻⁴. Similar discoveries of naked AGNs, characterized by the absence of a BLR accompanied by strong continuum emission and strong variability in the optical band and/or without significant intrinsic absorption in X-rays, indicate that these sources genuinely lack a BLR (see Hawkins 2004; Gliozzi et al. 2007).

Observations of Sy2s also support the existence of "true Seyfert 2" galaxies. Of the brightest Sy2s only ∼50% exhibit the presence of hidden BLRs in their polarized optical spectra (Tran 2001, 2003; Moran 2007). The luminosities of Sy2s without a hidden BLR have systematically lower luminosity/Eddington ratios (Lumsden & Alexander 2001; Gu & Huang 2002; Tran 2001, 2003; Moran 2007; Bian & Gu 2007) than Sy2s with the hidden BLR (obscured by inclination effects). This implies that "true Seyfert 2s" exist at low luminosities. Quantitative investigations of the non-BLR yield an estimate for the threshold of the Eddington ratio of 10^−1.37 for Sy2s (Bian & Gu 2007) and 10⁻² for naked AGNs (Gliozzi et al. 2007), both of which are higher than the theoretical prediction. We note, however, that large uncertainties exist in estimating the bolometric luminosity from the optical or X-ray luminosity (e.g., the bolometric luminosity in Bian & Gu 2007 should be 1.5 dex lower if based on the method used in Gliozzi et al. 2007), and uncertainties are present in the determination of the black hole mass, both contributing to apparent deviations from the theoretical prediction.

As the evaporation is saturated to the maximal rate at high accretion rates (Eddington ratios), the accretion flow in the corona cannot increase with the accretion rate after a transition to the soft state. Thus, the accretion flow within the disk (rather than corona) increases with the increasing accretion rate. With efficient Compton scattering, the corona component could be even weaker at high accretion rates because the increased disk radiation leads to strong Compton cooling, resulting in the condensation of coronal gas and the reduction of the optical depth in the corona. If we adopt the maximal evaporation rate without Compton cooling as an upper limit for the corona flow, we obtain an upper limit on the ratio of the mass inflow rates through the corona and disk as

$$\begin{equation} {\dot{m}_c\over \dot{m}_d}={1\over {\dot{m}\over 0.027}-1}, \end{equation} \tag{ 14 }$$

which is also an upper limit to the ratio of the coronal luminosity to the disk luminosity of a source in the high/soft state. For an accretion rate of 0.1, the coronal luminosity is no more than 37% of the disk luminosity and is less than ∼3% of the disk luminosity if the system accretes at the Eddington rate. This implies that the spectral index between optical and X-rays, α_ox, is very large in luminous AGNs.

However, observations reveal that the X-ray luminosity is comparable with the optical luminosity in some HLAGNs, indicating a strong corona in AGNs (which is similar to the very high state in BHXRBs). To produce such strong X-ray radiation in luminous AGNs, the frictional heating (in the model) is insufficient. To address this conundrum, it has been suggested that a large fraction of accretion energy is transported from the disk to the corona, perhaps due to a magnetic buoyant instability (Liu et al. 2002b; 2003; Cao 2009). Alternatively, the viscosity parameter α in these HLAGNs may be large, supporting a strong corona by means of efficient evaporation. Such a possibility could be a consequence of a larger magnetic Prandtl number in AGNs than in BHXRBs (see Lesur & Longaretti 2007).

In addition to their apparent strong coronae, AGNs exhibit other observational features that differ from BHXRBs. For example, among the luminous AGNs, ∼10% are radio-loud quasi-stellar objects (QSOs), while jets in BHXRBs appear to quench whenever the objects transit to the high/soft state. This difference may reflect that jets in the high state of BHXRBs are weak or the properties of the magnetic field in the corona, related to the magnetic Prandtl number, are modified (see above) to facilitate jet formation in AGNs.

We also note that soft X-ray excesses are seen in AGNs at high Eddington ratios, but not in BHXRBs. Such excesses appear to be correlated with the presence of the big blue bump. The origin of this excess is unknown, but it has been interpreted in terms of Comptonization in the disk (Kawaguchi et al. 2001) or resulting from a depression of the continuum at X-ray energies in the range 0.7–5 keV by relativistically broadened line transitions of O vii and O viii (Gierlinski & Done 2004). Its restriction to AGNs remains to be clarified.

The environment of a galactic central region can significantly differ from that for a stellar mass black hole in an X-ray binary system. For example, significant high temperature diffuse gas has been observed in the Galactic center, with Sgr A* accreting at very low Eddington ratios, which may also be present in some LLAGNs. Such hot gas would affect the boundary condition on the disk. However, in the case of hot gas joining the evaporating gas in the corona, the corona is not changed significantly. This is a consequence of the fact that the evaporation of the disk is depressed by the directly inflowing gas due to the pressure and energy balance between the disk and the corona. If the density in the corona is too high, the efficient cooling to the corona can even lead to the coronal gas condensing onto the disk. Consequently, the corona flow remains similar to the case of evaporation-fed corona. In other words, for a given accretion rate, independent of whether the disk is fed at the outer boundary, or the disk (corona) is partially fed in the form of cold (hot) gas, the accretion flows through the disk and through the corona in regions of a few hundred Schwarzschild radii are expected to be similar.

Finally, the disk instability model may not give rise to large amplitude outbursts (and hence transitions between the low/hard state and the high/soft state) due to the ionization instability in AGNs as in BHXRBs because the α parameter in the cold state may not significantly differ from its value in the hot state (see Menou & Quataert 2001). In this case, the variability would be of small amplitude and state transitions would be less dramatic. AGNs would therefore primarily reflect the mass inflow rate from its surrounding region rather than a temporal variation in a non-steady disk. On the other hand, gravitational instabilities may operate in AGNs to give rise to more significant time-dependent accretion.

In the future, we plan to carry out additional investigations to quantitatively confront the disk corona model with the detailed observations of AGNs, with comparisons based on a two-temperature corona model (Liu et al. 2002a) incorporating the effect of magnetic fields (Qian et al. 2007) and viscosity (Qiao & Liu 2009). In addition, a study of their differences with respect to BHXRBs will be explored to examine whether they are one of degree rather than of kind.