THE STRUCTURE AND SPECTRAL FEATURES OF A THIN DISK AND EVAPORATION-FED CORONA IN HIGH-LUMINOSITY ACTIVE GALACTIC NUCLEI

The disk evaporation model was proposed by Meyer & Meyer-Hofmeister (1994) for dwarf novae and developed for black hole binaries by Meyer et al. (2000a). In this model, an optically thin, hot corona is assumed to lie above a thin disk. Vertical conduction is important since the disk temperature is much lower than that of the corona. As the corona radiates inefficiently, the conductive heat accumulates downward and at the transition layer it heats up some cool gas until an equilibrium density is reached so as to completely radiate the conductive flux. The evaporated gas is then accreted through the corona to the central black hole. Such a corona is heated up by viscosity and the accretion is supplied by mass evaporation from the disk. Further investigations on the effects of decoupling between ions and electrons and Compton cooling in the corona (Liu et al. 2002a), the magnetic pressure (Meyer-Hofmeister & Meyer 2001; Qian et al. 2007), the change of the viscosity parameter (Qiao & Liu 2009), and the possible condensation are included. The model has been successfully used to interpret the state transition and observational features in the low/hard state of the black hole binaries. As the disk corona is independent of the black hole mass, it is also applied to AGNs in explaining different types of spectra and the absence of the broad line region in LLAGNs (Liu & Taam 2009). Nevertheless, when it is compared with observations in HLAGN, we find that in the corona, strong Compton cooling causes the corona to be so weak that it cannot produce the hard X-ray as high as observed. There must be other energy sources for the corona besides viscous heating.

There are some pioneer works assuming that a certain fraction of local gravitational energy is liberated directly in the corona (e.g., Nakamura & Osaki 1993; Haardt & Maraschi 1991, 1993), particularly in the frame of a magnetized disk corona (e.g., Di Matteo 1998; Di Matteo et al. 1999; Miller & Stone 2000; Merloni & Fabian 2002; Liu et al. 2002b, 2003; Kawanaka et al. 2008; Cao 2009). In these investigations, it is demonstrated that the corona can indeed produce as strong an X-ray as is observed in HLAGNs if a large fraction of accretion energy is released in the corona. Nevertheless, the mass exchange between the two-phase accretion flows was not taken into account. Here, we consider both radiation and mass coupling between disk and corona and study the detailed vertical structure of the coronal flow above a thin disk, supposing some fraction ( f ) of the gravitational energy is added into the corona.

The corona can be described by the following equations. The equation of state is

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

where μ = 0.62 is the molecular weight assuming a standard chemical composition (X = 0.75, Y = 0.25) for the corona. Here, we assume n_i = n_e, which is strictly true only for a pure hydrogen plasma.

The equation of continuity is

$$\begin{equation} \centering {d\over dz}(\rho v_{\rm z})=\eta _{\rm M}{2\over R}\rho v_{\rm R} -{2z\over R^2+z^2}\rho v_{\rm z}, \end{equation} \tag{ 2 }$$

where v_R = −α(V²_s/ΩR) is the radial component of velocity, V_s = (P/ρ)^1/2 is the isothermal sound speed, and $v_{\rm \varphi }=\sqrt{({\it G M}/R)} (1+({z^2/R^2}))^{-4/3}$ is the angular component of velocity. Here, the partial derivative of mass flux with respect to radius is approximated as, (1/R)(∂Rρv_R/∂R) ≈ −η_M(2/R)ρv_R, where η_M depends on the net mass gain/loss rate through the radial boundaries (Meyer-Hofmeister & Meyer 2003; Liu et al. 2004). In this work, η_M = 1 is taken for the case that no mass enters through the outer boundary and all the mass flowing in the corona is contributed by the evaporation (Meyer et al. 2000a).

The equation of the z-component of momentum is

$$\begin{equation} \rho v_{\rm z} {dv_{\rm z}\over dz}=-{{\it dP}\over dz}-\rho {{\it GM}z\over (R^2+z^2)^{3/2}}. \end{equation} \tag{ 3 }$$

In the hot corona, because of a heavier mass of ions than that of electrons, the viscous heating raises the ion temperature first and ions are cooled by coulomb collision with electrons and both radial and vertical advection. Here, we do not include direct heat to electrons. The ion thermal conduction is not taken into account because of the long mean free path compared to the vertical scale height of the corona. So the energy equation for ions is

$$\begin{eqnarray} &&\frac{d}{dz}\left\lbrace \rho _{\rm i} v_{\rm z} \left[{v^2\over 2}+{\gamma \over \gamma -1}{P_{\rm i}\over \rho _{\rm i}}-{{\it GM}\over (R^2+z^2)^{1\over 2}}\right]\right\rbrace ={3\over 2}\alpha P\Omega -q_{\rm ie}\nonumber\\ &&\quad+{\eta _{\rm E}}{2\over R}\rho _{\rm i} v_{\rm R} \left[{v^2\over 2}+{\gamma \over \gamma -1}{P_{\rm i}\over \rho _{\rm i}}-{{\it GM}\over (R^2+z^2)^{1\over 2}}\right]\nonumber\\ &&\quad-{2z\over {R^2+z^2}}\left\lbrace \rho _{\rm i} v_{\rm z} \left[{v^2\over 2}+{\gamma \over \gamma -1}{P_{\rm i}\over \rho _{\rm i}}-{{\it GM}\over (R^2+z^2)^{1\over 2}}\right]\right\rbrace, \end{eqnarray} \tag{ 4 }$$

where η_E is the energy modification parameter, η_E = η_M + 0.5. The difference between η_E and η_M comes from the derivative of energy flux with respect to radius, (1/R)(∂Rρv_R/∂R) ≈ −η_M(2/R)ρv_R + ρv_R(∂/∂R) = −(η_M  +  0.5)(2/R)ρv_R, where ≡ v²/2 + (γ/γ − 1)(P_i/ρ_i) − GM/(R² + z²)^1/2 and its derivative (∂/∂R) ≈ −(/R) since the potential, kinetic, and thermal specific energies all scale approximately as 1/R (Meyer-Hofmeister & Meyer 2003; Liu et al. 2004). In Equation (4), q_ie is the exchange rate of energy between electrons and ions through coulomb collision and is described as

$$\begin{equation} {q_{\rm ie}}={\bigg ({2\over \pi }\bigg)}^{1\over 2}{3\over 2}{m_{\rm e}\over m_{\rm i}}{\ln \Lambda }{\sigma _{\rm T} c n_{\rm e} n_{\rm i}}(\kappa T_{\rm i}-\kappa T_{\rm e}) {{1+{T_{\rm *}}^{1\over 2}}\over {{T_{\rm *}}^{3\over 2}}}, \end{equation} \tag{ 5 }$$

in which

$$\begin{equation} T_{\rm *}={{\kappa T_{\rm e}}\over {m_{\rm e} c^2}}\bigg (1+{m_{\rm e}\over m_{\rm i}}{T_{\rm i}\over T_{\rm e}}\bigg), \end{equation} \tag{ 6 }$$

where m_i and m_e are the proton and electron masses, κ is the Boltzmann constant, c is the light speed, σ_T is the Thomson scattering cross section, and ln Λ = 20 is the Coulomb logarithm.

Different from the previous works, besides the viscous heating, a fraction of local gravitational energy is assumed to directly heat the electrons in corona, i.e.,

$$\begin{equation} Q_{\rm add}=\frac{3{\it GM}\dot{M}}{8\pi R^3}[1-(3\,R_{\rm s}/R)^{1/2}]\times f, \end{equation} \tag{ 7 }$$

where $\dot{M}$ is the total accretion rate in the disk corona. The physical meaning of this additional heating term will be discussed in Section 4. To simplify the calculations, this additional heating flux is supposed to distribute along the vertical direction in a form similar to that of Compton cooling, i.e.,

$$\begin{equation} q_{\rm add}={\frac{4\kappa T_{\rm e}}{m_{\rm e} c^2}}n_{\rm e} \sigma _{\rm T} c {\frac{a T_{\rm eff0} ^4}{2}}, \end{equation} \tag{ 8 }$$

where

$$\begin{equation} T_{\rm eff0}=\left\lbrace \frac{3{\it GM}\dot{M}}{8\pi R^3 \sigma }[1-(3\,R_{\rm s}/R)^{1/2}]\times \lambda \right\rbrace ^{\frac{1}{4}} \end{equation} \tag{ 9 }$$

and λ is determined by the integration along z from the lower boundary to the upper boundary, $\int _{z_0}^{z_1}q_{\rm add}dz=Q_{\rm add}$, which gives the relation of f and λ,

$$\begin{equation} f=2\lambda \int _{z_0}^{z_1}{\frac{4\kappa T_{\rm e}}{m_{\rm e} c^2}}n_{\rm e} \sigma _{\rm T}dz. \end{equation} \tag{ 10 }$$

Therefore, the energy equation for both the ions and electrons is

$$\begin{eqnarray} &&\frac{d}{dz}\left\lbrace \rho {v}_{\rm z}\left[{v^2\over 2}+{\gamma \over \gamma -1}{P\over \rho } -{{\it GM}\over (R^2+z^2)^{1/2}}\right] + F_{\rm c} \right\rbrace \nonumber\\ &&\ \ \ =\frac{3}{2}\alpha P{\Omega }+q_{\rm add}-n_{\rm e}n_{\rm i}L(T_e)-q_{\rm Cmp}\nonumber\\ &&\quad\ \ \ +\eta _{\rm E}{2\over R}\rho v_{\rm R} \left[{v^2\over 2}+{\gamma \over \gamma -1}{P\over \rho } -{{\it GM}\over (R^2+z^2)^{1/2}}\right]\nonumber\\ &&\quad\ \ \ -{2z\over R^2+z^2}\left\lbrace \rho v_{\rm z}\left[{v^2\over 2}+{\gamma \over \gamma -1}{P\over \rho }- {{\it GM}\over (R^2+z^2)^{1/2}}\right] +F_{\rm c}\right\rbrace. \nonumber\\ \end{eqnarray} \tag{ 11 }$$

In this equation, n_en_iL(T_e) is the bremsstrahlung cooling rate and F_c is the thermal conduction (Spitzer 1962),

$$\begin{equation} F_{\rm c}=-\kappa _{\rm 0}T_{\rm e}^{5\over 2}{dT_{\rm e}\over dz}, \end{equation} \tag{ 12 }$$

with $\kappa _0 = 10^{-6}\,{\rm erg\,s^{-1}\,cm^{-1}\,K^{-{7\over 2}}}$ for fully ionized plasma.

For the Compton cooling rate, q_Cmp, we mainly consider inverse Compton scattering of the soft photons from the disk, while the ones contributed from the bremsstrahlung cooling are not included:

$$\begin{equation} q_{\rm Cmp} = {\frac{4\kappa T_{\rm e}}{m_{\rm e} c^2}}n_{\rm e} \sigma _{\rm T} c {\frac{a T_{\rm eff} ^4}{2}}, \end{equation} \tag{ 13 }$$

where a is the radiation constant and T_eff is the effective temperature of the underlying thin disk, fulfilling

$$\begin{equation} \sigma T_{\rm eff}^4=\frac{3{\it GM}(\dot{M}_{\rm d}-f\dot{M})}{8\pi R^3 }[1-(3\,R_{\rm s}/R)^{1/2}], \end{equation} \tag{ 14 }$$

where $\dot{M}_{\rm d}$ is the mass accretion rate in the thin disk, which depends on the distance because of evaporation,

$$\begin{equation} \dot{M}_{\rm d} (R)= \dot{M}-\dot{M}_{\rm evap} (R), \end{equation} \tag{ 15 }$$

with $\dot{M}_{\rm evap}(R)$ as the integrated evaporation rate from the outer edge of the disk to the distance R, which can be approximated by the one-zone evaporation rate, $\dot{M}_{\rm evap}=2\pi R^2(\rho v_{\rm z})_{0}$ (see Liu et al. 2002a). The negative term associated with $-f\dot{M}$ in Equation (14) corresponds to the transportation of accretion energy from the disk to the corona, which results in a lower temperature in the disk. Here, we have not considered back reaction (heating of a thin disk by coronal illumination) and its justification will be discussed later.

These five differential equations, Equations (2), (3), (4), (11), and (12), which contain five variables P(z), T_i(z), T_e(z), F_c(z), and ρv_z, can be solved with five boundary conditions.

At the lower boundary, the temperature of the gas should be the effective temperature of the accretion disk. Liu et al. (1995) showed that the coronal temperature increases from effective temperature to 10^6.5 K in a very thin layer of nearly constant pressure. Its physics is described by the balance between thermal conduction and radiation loss. So a relation can be established between temperature and heat flux which can be scaled according to the pressure. Combining with Shmeleva–Syrovatskii relation as F_c = −((κ₀L(T)T^3/2)^1/2/κ)p (Shmeleva & Syrovatskii 1973), the lower boundary conditions can be approximated (Meyer et al. 2000a) as

$$\begin{equation} \begin{array}{l}T_{\rm i}=T_{\rm e}=10^{6.5} \,{\rm K},\\ F_{\rm c}=-2.73\times 10^6 P \ {\rm at}\ z=z_0. \end{array} \end{equation} \tag{ 16 }$$

There is no pressure and no heat flux at infinity, which requires sound transition at some height z = z₁. So we constrain the upper boundary as

$$\begin{equation} F_{\rm c}=0\ {\rm and} \ v_{\rm z}=V_{\rm s}\ {\rm at}\ z=z_1. \end{equation} \tag{ 17 }$$

With such boundary conditions, assuming an initial value λ and a pair of lower boundary values for P₀ and (ρv_z)₀, we start the integration along z. If the trial values for P₀ and (ρv_z)₀ fulfill the upper boundary conditions, then the initial value of P₀ and (ρv_z)₀ can be taken as the solutions of the differential equations. The value of f can then be calculated out from Equation (10). The results for a series of f = 0.0, 0.1, 0.2, 0.3, 0.4 are given in the following section. When f = 0.0, which means that there is no other energy heating the corona except the viscous heating, this result is similar to the former works and will be compared with the results for f > 0.0.

With the detailed calculations of the corona structure, the temperature and density in the corona and the effective temperature are determined. We then calculate the spectrum from such a disk and corona for a given structure.

In the corona, the emissions are from bremsstrahlung radiation and inverse Compton scattering of disk photons. For the bremsstrahlung cooling spectrum, we adopt the Eddington approximation and two-stream approximation (Rybicki & Lightman 1979; Manmoto et al. 1997) to calculate the radiative flux,

$$\begin{equation} F_{\rm \nu }=\frac{2\pi }{\sqrt{3}}B_{\rm \nu }[1-\rm {exp}(-2\sqrt{3}\tau ^{*}_{\rm \nu })], \end{equation} \tag{ 18 }$$

where B_ν is the Planck function and $\tau ^{*}_{\rm \nu }=\sqrt{\pi }\kappa _{\rm \nu }H$ is the vertical absorption optical depth. Assuming that the flows are local thermal equilibrium, then κ_ν = χ_ν/4πB_ν, where χ_ν = χ_{ν, brems} is the emissivity of bremsstrahlung (Narayan & Yi 1995b).

Since we focus on HLAGNs, for which the accretion rate is very high and the disk is the dominant accretion flow, the soft photons for inverse Compton scatter are supplied by the thin disk blackbody radiation. However, the ones from bremsstrahlung emission are neglected because they have similar energies to the electrons and cannot gain much from Compton scattering (Done 2010). We compute the Compton spectrum by a Monte Carlo simulation. The accretion flow consists of a cold disk in the middle and a hot corona above/below. At each radius, the cold disk emits blackbody radiation with the temperature T_eff as shown in the model Equation (14). The soft photons from the local disk move transverse through the hot corona. Since the hot corona is optically thin, some of the soft photons pass through it without scattering, and some of them are scattered by the electrons in the corona. We calculate the energy spectra of photons that emerge from the corona. The method of Monte Carlo simulation used in our paper is essentially the same as that described by Pozdniakov et al. (1977). In the code, we introduce the weight ω to efficiently calculate the effects of multiple scattering. First, we set the initial weight ω₀ = 1 for a given soft photon from the cold disk. Then, we calculated the escape probability P₀ of passing through the corona slab. The value of ω₀P₀ is the transmitted portion and is recorded to calculate the penetrate spectrum. The remaining weight ω₁ = ω₀(1 − P₀) is the portion that undergoes more than one scattering. Suppose that P_n is the escape probability after the nth scattering, then ω_nP_n is the transmitted portion of photons after the nth scattering. The remaining portion ω_n(1 − P_n) undergoes the (n + 1)th scattering. This calculation is continued until the weight ω_n becomes sufficiently small. In our calculation, the number of processes calculated is 100,000 and ω_n < 10⁻⁴. Finally, we obtain the transmitted spectrum and the soft spectrum.

First, we set λ = 0.0, meaning f = 0, which is the case when the corona is heated by only viscous friction. We calculate the corona structures at R = 1000 R_s for $\dot{m}=0.02$ and $\dot{m}=0.2$ as shown in the Figure 1; here, $\dot{m}=\dot{M}/\dot{M}_{\rm Edd}$, which is the total accretion rate. Comparing the results in two panels, we find that the vertical profiles of the coronal quantities are similar for different accretion rates. From the lower boundary upward, the pressure (P) and the vertical mass flow (ρv_z) decrease dramatically in a thin layer. The downward heat flux (−F_c) increases in this layer until a maximum is reached and then decreases along z and can be neglected on the upper boundary. The temperatures of electrons (T_e) and ions (T_i), starting at a coupled value of 10^6.5 K at the coronal lower layer, increase with height and decouple at about z/R ∼ 0.1. Upon this height, T_i still increases up to T_i ∼ 0.3 T_vir and T_e keeps almost the same value (∼0.05 T_vir) throughout the corona, where T_vir = GMm_pμ/κR = 3.37 × 10¹²(R/R_s)⁻¹ K is the virial temperature. These typical features in disk corona, which are also shown in earlier works (e.g., Meyer et al. 2000b), indicate that the corona above a thin disk undergoes very steep changes in temperature and density and cannot be simply averaged in the vertical direction.

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We also calculate the evaporation rate $\dot{m}_{\rm evap}$ ($=\dot{M}_{\rm evap}/\dot{M}_{\rm Edd}$) along distances, which represents the mass flowing rate in the corona. The results are shown in Figure 2. Similar to the previous works (e.g., Liu at al. 2002a), the coronal accretion rate increases toward the central black hole, reaches a maximum at a few hundred Schwarzschild radii, and then drops very quickly in the inner region because of strong inverse Compton scattering, i.e., a large fraction of the corona gas condensing onto the disk near the black hole. Comparing the three evaporation curves for accretion rate, $\dot{m}$ = 0.02, 0.2, and 0.4, we find that the maximum evaporation rate decreases with increasing accretion rate and the corona gas begins condensing at outer regions for higher accretion rate. The temperature in electrons, as shown in Figure 1, is lower at higher accretion rates. This indicates that the corona becomes very weak at high accretion rates, like in HLAGNs.

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We can roughly estimate the radiation from the disk and corona under viscous heating. As shown in Figure 2, the maximum corona accretion rate is $\dot{m}_{\rm evap,max}=4\times 10^{-3}$ at ∼1250 R_s for $\dot{m}=0.2$. Supposing that the corona keeps accreting at this accretion rate, the radiation strength of the corona relative to the disk is

$$\begin{equation} \frac{L_{\rm c}}{L_{\rm d}}= \frac{\eta \dot{M}_{\rm c}c^2}{\eta \dot{M}_{\rm d}c^2} \lesssim \frac{\dot{m}_{\rm evap,max}}{\dot{m}-\dot{m}_{\rm evap,max}}=0.02. \end{equation} \tag{ 19 }$$

Here, we have supposed that radiative efficiency η in the corona is equal to that in the disk. From Equation (19), we can see that the corona is very weak and the radiation is dominated by the disk. In fact, the luminosity ratio between the corona and disk is much smaller than the above value since more and more coronal gas condenses into the disk at smaller distance (R < 1250 R_s). In order to produce strong X-ray in the corona, we consider that some fraction of gravitational energy is transferred to the corona, providing an additional heating mechanism against the strong Compton cooling.

3.2.1. Vertical Structure

Assuming a fraction f (=0.1, 0.2, 0.3, and 0.4) of accretion energy is added to the corona, we perform numerical calculations for accretion rates of $\dot{m}=$ 0.1, 0.2, and 0.5.

We start our calculations from a distance of 100 R_s to the ISCO region (R = 3.1 R_s for convenience). It has been shown in Section 3.1 that the maximum corona accretion rate can only reach ${\sim} 4\times 10^{-3}\,\dot{M}_{\rm Edd}$ at ∼1000 R_s for $\dot{m}=0.2$. In the inner region, strong Compton cooling leads to condensation of the corona. Whether the corona can exist at distances inside 100R_s depends on how much accretion energy is added to the corona. From our test calculations, we find that for f = 0.2, a corona can still survive in the region from 100 R_s to 3.1 R_s. The coronal vertical structure is shown in Figure 3. The profile is similar to the case for f = 0 at large distances, though the absolute values are different.

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In order to study the corona properties with different energy input, we calculate the coronal accretion rate for f = 0.1, 0.2, 0.3, 0.4, where the total mass supply rates are assumed to be $\dot{m}=0.2, 0.5,$ respectively. The results are plotted in Figures 4 and 5. One can see from Figure 4 that the accretion rate in the corona decreases with decreasing distance as a consequence of condensation, while the electron temperature and scattering optical depth are nearly the same at distances up to 100 R_s, as shown in Figure 5. Besides, Figure 4 also shows that the larger the fraction f is, the larger the gas flow becomes in the corona. This is because more energy goes directly to heat the electrons in the corona to a higher temperature, leading to more conductive flux and hence more evaporation or less condensation. Thus, more gas is kept in and accreted through the corona. This is confirmed by the higher electron temperature and larger scattering optical depth for larger f, as shown in Figure 5.

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Comparing the two panels in Figure 4, we understand that the coronal accretion rates are insensitive to the mass supply rate as long as the fraction of accretion energy released in the corona is the same. Calculations show that the electron temperature and scattering optical depth of the corona do not vary with the accretion rate either. This can be understood as a consequence of energy balance in the corona. For a given f, when the mass supply rate ($\dot{m}$) increases, the heating rate added to the corona also increases (${\propto} f\dot{m}$). On the other hand, the cooling rate by inverse Compton scattering also increases with the increasing density of soft photon (${\propto} (1-f)\dot{m}$, if $(1-f)\dot{m} \gg \dot{m}_{\rm evap}$). If the additional heat and Compton cooling are the dominant energy terms for the coronal electrons, then there should be no variation in both the evaporation rate and Compton y-parameter. Therefore, the electron temperature does not vary either.

3.2.2. Spectra of the Disk and Corona

Deriving the structure of our model at each radius, we get the radius-dependent electron temperature and electron scattering optical depth, which are the input parameters for bremsstrahlung and Compton spectrum. The bremsstrahlung radiation is proportional to n²_e but the inverse Compton scattering is proportional to n_e. So from the vertical structure, we can see that the bremsstrahlung dominates in the lower layer of the corona and the upper layer is dominated by the Compton radiation. Since the temperature and number density in the upper corona still depend on the vertical height, we need to choose an appropriate temperature and density at every given distance for calculating the Compton spectrum with a Monte Carlo simulation. Here, we constrain them by the luminosity, i.e., when the luminosity derived from the Monte Carlo simulation equals the one derived from the coronal structure calculation, the chosen temperature and density are thought to be correct and we take this spectrum as the true spectrum from the corona.

Figure 6 shows the spectra for different f, f = 0.1, 0.2, 0.3, 0.4, and different accretion rates, $\dot{m}=0.2, 0.5$. The spectrum is Compton dominated in the high accretion rate. The larger the f is, the stronger both the Compton and the bremsstrahlung radiation become, since the coronal density and temperature increase with increased heating to the corona; while the optical-UV radiation, which is from the disk, slightly decreases with increasing f. The spectrum of hard X-ray in the range of 2–10 keV is marked by the two vertical dashed lines. The hard X-ray spectrum is harder for higher f and L_bol/L_{2 − 10 keV} decreases with f. For f = 0.1, the spectrum seems so steep that the corona radiation is still very weak. The photon index for the hard X-ray is in the range of 2.2–2.7 for 0.2 ⩽ f ⩽ 0.4, which roughly fits the spectrum of HLAGNs. Table 1 lists the photon index for 2–10 keV and the ratio of L_bol/L_{2 − 10 keV}. For the different accretion rates, the spectrum index is nearly the same with the same f, which can also be seen from the total spectrum shown in Figure 7. Comparing the two panels in Figure 7, we understand that the spectrum varies with the fraction of accretion energy liberating in the corona, while it does not sensitively depend on $\dot{m}$, as long as f is independent on the accretion rate. This can be understood from our structure calculating results; namely, for a given f, the electron temperature in the corona, T_e, and the scattering optical depth, τ, do not change as the accretion rate, though the disk radiation increases and hence Compton radiation also increases with increasing accretion rate.

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Table 1. The Photon Index and L_bol/L_{2 − 10 keV} for Different Accretion Rates with Different f

	$\dot{m}=0.1$		$\dot{m}=0.2$		$\dot{m}=0.5$
f	Γ	Lbol/L2 − 10 keV	Γ	Lbol/L2 − 10 keV	Γ	Lbol/L2 − 10 keV
0.2	2.6	228	2.6	222	2.7	222
0.3	2.3	41	2.3	39	2.3	36
0.4	2.2	30	2.2	29	2.2	28

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The photon index and the luminosity ratio, as listed in Table 1, indicate that a fraction of accretion energy is necessary in order to explain observations in HLAGNs. For 0.2 ⩽ f ⩽ 0.4, the predicted photon index is around 2.2–2.7, and the bolometric correction from hard X-rays L_bol/L_{2 − 10 keV} is in the range of 28–228. These are roughly consistent with observations (Vasudevan & Fabian 2007, 2009; Shemmer et al. 2006 and references therein; Zhou & Zhao 2010). In Figure 8, we show how the photon index in X-rays varies with bolometric luminosity. The figure reveals that the bolometric correction is larger for a steeper X-ray spectrum, which seems to be a common feature for objects with different Eddington ratios. This is a consequence of the decrease of energy fraction released in the corona. We now raise the questions, what are the underlying physics driving the accretion energy to release in the corona? Does the energy fraction f depend on the accretion rate or the black hole mass? We discuss this in the following section.

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Studying the magnetic field in the accretion flow has a long history, either with magnetohydrodynamics (MHD) simulations (e.g., Balbus & Hawley 1998; Matsumoto 1999; Stone & Pringle 2001; Machida et al. 2001; Machida & Matsumoto 2003; Kato et al. 2004; Kawanaka et al. 2008 and references therein; Ohsuga et al. 2009; Ohsuga & Mineshige 2011; Penna et al. 2010) or numerical simulations to interpret observations (Liu et al. 2002b, 2003; Cao 2009).

There are good reasons to believe that the magnetic field plays an important role in transporting the accretion energy from the disk to the corona (Haardt & Maraschi 1991). Liu et al. (2002b, 2003) investigated the Parker instability in the accretion flow, following the model for the solar corona (e.g., Yokoyama & Shibata 2001), and found that the accretion energy can be transported to the corona and heats the electrons in the corona. In the frame of disk corona with mass evaporation, Qian et al. (2007) analyzed the coronal structure and evaporation features with the magnetic field and found that the maximum evaporation rate stays more or less constant (${\sim} 3\% \,\dot{M}_{\rm Edd}$) when the strength of the magnetic field changes, but that the radius corresponding to the maximum evaporation rate decreases with increasing magnetic field. Qiao & Liu (2009) investigated the influence of a viscosity parameter α in the corona and suggested that the coronal radiation is largely enhanced when viscosity increases.

However, it is not very clear how the magnetic field connects with viscosity in the evaporation-fed corona, though some investigations of accretion flows show that the magnetic field and viscosity are associated (e.g., Balbus & Hawley 1991; Matsumoto & Tajima 1995; Abramowicz et al. 1996; Balbus 2003). Here, we discuss whether the predicted spectrum can be consistent with observations if it is the magnetic field that brings a fraction of gravitational energy to the corona.

Let us assume that the magnetic field is generated in the disk by dynamo action. Then, the magnetic loops can emerge out of the disk by Parker buoyant instability. In this process, the accretion energy stored in the magnetic field is transported into the corona and thereby liberated through magnetic reconnection. According to the equipartition theorem between the magnetic pressure and gas pressure, we set P_B = B²/8π = (1/β)P_{g, d}, where P_B and P_{g, d} are magnetic pressure and gas pressure in the disk. Then, the magnetic energy flux is

$$\begin{equation} F_{\rm B}=\frac{B^2}{4\pi }\sqrt{\frac{B^2}{4\pi \rho _{\rm d}}}=(2P_{\rm B})^{\frac{3}{2}}\rho _{\rm d}^{-\frac{1}{2}}=\left(\frac{2}{\beta }P_{\rm g,d}\right)^{\frac{3}{2}} \rho _{\rm d}^{-\frac{1}{2}}. \end{equation} \tag{ 20 }$$

If this part of the magnetic energy flux is the additional heating source for the corona, then the ratio of this part of energy to total gravitational energy is

$$\begin{equation} f=\frac{F_{\rm B}}{F_{\rm tot}}=\left(\frac{2}{\beta }P_{\rm g,d}\right)^{\frac{3}{2}}\rho _{\rm d}^{-\frac{1}{2}}\bigg/F_{\rm tot}, \end{equation} \tag{ 21 }$$

where F_tot is the total accretion flux through the accretion process, $F_{\rm tot}=({3{\it GM}\dot{M}}/{8\pi R^3 })[1-(3\,R_{\rm s}/R)^{1/2}]=8.56\times 10^{26}\, \dot{m}m^{-1}\, r^{-3}\Phi$, where Φ = 1 − (3/r)^1/2.

In the thin disk, the pressure of the disk is

$$\begin{equation} P=P_{\rm g,d}+P_{\rm B}+P_{\rm r,d}=\left(1+\frac{1}{\beta }\right)\frac{\rho _{\rm d} k T_{\rm d}}{\mu m_{\rm p}}+\frac{aT_{\rm d}^4}{3}, \end{equation} \tag{ 22 }$$

where μ = 0.62 is the averaged molecular weight. Since some fraction ( f ) of gravitational energy is carried into the corona by magnetic activity, the energy equation in the disk is

$$\begin{equation} \frac{4ac T_{\rm d}^4}{3\tau }=(1-f)\times F_{\rm tot}=(1-f)\frac{3{\it GM}\dot{M}}{8\pi R^3 }[1-(3\,R_{\rm s}/R)^{1/2}], \end{equation} \tag{ 23 }$$

where τ = (κ_es + κ₀ρ_dT^−3.5_d)Σ, κ_es = 0.4 cm² g⁻¹, κ₀ = 6.4 × 10²² cm² g⁻¹, and Σ = 2ρ_dH_d is the surface density of the disk. The angular momentum equation in the disk is

$$\begin{equation} \frac{\dot{M}}{3\pi }\Phi =\nu \Sigma, \end{equation} \tag{ 24 }$$

where ν = (2/3)αc_sH_d, the viscous velocity c_s = ΩH_d.

If the gas pressure in the disk is dominant over radiation pressure, P_{g, d} ≫ P_{r, d}, and Thomson scattering is much higher than free–free absorption, σ_T ≫ σ_ff, then we have τ ≈ κ_esΣ, which is similar to case (b) in Shakura & Sunyaev (1973), except that r = R/R_s is taken here instead of r = R/3 R_s. We can derive the disk quantities. The density in the disk is

$$\begin{equation} \rho _{\rm d}=21.43(\alpha m)^{-\frac{7}{10}}[\dot{m}\Phi]^{\frac{2}{5}}r^{-\frac{33}{20}}\left(1+\frac{1}{\beta }\right)^{-\frac{6}{5}}(1-f)^{-\frac{3}{10}}, \end{equation} \tag{ 25 }$$

and the gas pressure is

$$\begin{equation} P_{\rm g,d}=1.91\times 10^{18}(\alpha m)^{-\frac{9}{10}}[\dot{m}\Phi]^{\frac{4}{5}}r^{-\frac{51}{20}}\left(1+\frac{1}{\beta }\right)^{-\frac{7}{5}}(1-f)^{-\frac{1}{10}}. \end{equation} \tag{ 26 }$$

With this disk pressure and density, the energy fraction can be obtained from Equation (21),

$$\begin{equation} f=1.88\alpha ^{-1}(1+\beta)^{-\frac{3}{2}}. \end{equation} \tag{ 27 }$$

One can see that f is independent of the accretion rate if the disk is gas pressure dominated. This implies that a constant fraction of disk accretion energy is transported to the corona at different accretion rates if equipartition coefficient β does not vary with accretion rates. If this is the case for the additional heating to the corona, then the model spectrum will not vary with the accretion rate, as shown in Figure 7.

In fact, the accretion disk around a supermassive black hole is often dominated by radiation pressure. In the case of P_{g, d} ≪ P_{r, d} and σ_T ≫ σ_ff, the density and pressure in the disk are expressed as

$$\begin{equation} \rho _{\rm d}=1.14\times 10^{-6}(\alpha m)^{-1}[\dot{m}\Phi]^{-2}r^{\frac{3}{2}}(1-f)^{-3}, \end{equation} \tag{ 28 }$$

$$\begin{equation} P_{\rm g,d}=6.21\times 10^{9}(\alpha m)^{-\frac{5}{4}}[\dot{m}\Phi]^{-2}r^{\frac{9}{8}} (1-f)^{-\frac{13}{4}}, \end{equation} \tag{ 29 }$$

which are similar to the solution of case (a) in Shakura & Sunyaev (1973). Then, we can determine f from Equation (21) as

$$\begin{equation} f(1-f)^{\frac{27}{8}}=1.52\times 10^{-9}\alpha ^{-\frac{11}{8}}m^{-\frac{3}{8}}(\dot{m}\Phi)^{-3}r^{\frac{63}{16}}{\beta }^{-\frac{3}{2}}. \end{equation} \tag{ 30 }$$

Apparently, in a radiation-pressure-dominated disk, f depends on the accretion rate. The function f(1 − f)^27/8 in Equation (30) reaches a maximum of 0.095 at f = 8/35, which requires $1.52\times 10^{-9}\alpha ^{-{11}/{8}}m^{-{3}/{8}}(\dot{m}\Phi)^{-3}r^{{63}/{16}}{\beta }^{-{3}/{2}} \le 0.095$. This indicates that the accretion rate must be larger than a certain value in order to get a solution for f from Equation (30). If this condition $1.52\times 10^{-9}\alpha ^{-{11}/{8}}m^{-{3}/{8}}(\dot{m}\Phi)^{-3}r^{{63}/{16}}{\beta }^{-{3}/{2}} \le 0.095$ is fulfilled, then there are two sets of solutions for f (we saw this bimodal nature in Liu et al. 2002b). As shown in Figure 9, one set of solutions is in the range of 0 < f < (8/35), where f decreases with increasing accretion rate $\dot{m}$, while the other one is in the range of (8/35) < f < 1, where f increases with increasing $\dot{m}$. The relation between the energy fraction and the accretion rate, i.e., $f(\dot{m})$, depends on the viscosity and magnetic field. In the left panel of Figure 9, we show the effect of viscosity by assuming β = 1 under the theory of the equipartition of gas pressure and magnetic pressure. Actually, the local and global MHD simulation in the radiation-dominated accretion flow reveals that magnetic energy density may exceed gas energy density but be kept below radiation energy density, which means that β can be less than 1.0 (e.g., Turner et al. 2003; Ohsuga et al. 2009; Ohsuga & Mineshige 2011). The effect of magnetic fields is plotted in the right panel of Figure 9. It shows that the influence of the magnetic field is significant. The stronger the magnetic field, the larger $\dot{m}$ corresponding to f = 8/35. For a given accretion rate, for example, $\dot{m}=0.4$, f is larger with stronger magnetic field (i.e., smaller β) in the lower part of the curve (f < 8/35).

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Observed spectral features provide a clue to the possible range of f. If f < (8/35), then the fraction of accretion energy transferred to the corona decreases with accretion rate. As a consequence, the hard X-ray spectrum becomes softer at high accretion rates, as shown in Figure 10. However, if a large fraction of accretion energy (f > (8/35)) is transferred to the corona, f increases with accretion rate, consequently, the spectrum is harder at higher accretion rates. Comparing the model prediction and observations, a small fraction of accretion energy liberation in the corona is preferred, that is, <8/35. In Figure 10, we plot the spectrum for different accretion rates. The dashed lines represent a spectrum for a low accretion rate, and the dotted line is for a relatively higher accretion rate, and the solid line is for a much higher accretion rate, where the fraction of energy released in the corona corresponds to 0.3, 0.2, and 0.1, respectively, determined by Equation (30), assuming disk viscous parameter α = 0.01, β = 1 for an m = 10⁸ black hole at the most efficient radiation region r = 49/12. We show that the spectrum becomes steeper/softer at high accretion rates. Consequently, the bolometric correction from the 2–10 keV increases with increasing accretion rate.

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We point out that in the high accretion rate, the radiation-pressure-dominated disk is thermally and viscously unstable, which can lead to the S-shaped "limit cycle" (e.g., Shakura & Sunyaev 1976; Abramowicz et al. 1988; Honma et al. 1991; Szuszkiewicz & Miller 1998; see however, Hirose et al. (2009) who claimed no instability by local radiation-hydrodynamical simulations). However, if a fraction of accretion energy is liberated in the corona or the outflow, then the disk can be stable (Nakamura & Osaki 1993; Svensson & Zdziarski 1994). The magnetic heating to the corona discussed here is helpful to stabilize the disk, though f is not large enough.

In our corona, we assume that all of the viscously dissipated energy heats ions only. However, direct heating to the electrons can also increase the corona emission. According to the detailed modeling to Sgr A* (Yuan et al. 2003), a significant fraction of the viscously dissipated energy should heat electrons directly. Such a result was later supported by the numerical simulation done by Sharma et al. (2007). Nevertheless, this effect is negligible if much larger additional heating to the corona is assumed. When a strong disk exists under a corona as in luminous AGN, a small fraction of disk accretion energy transported to coronal electrons can dominate the coronal heating. For instance, assuming $\dot{m}=0.2$ and f = 0.1, the additional heating to the corona is 0.02 (in units of Eddington luminosity); while the accretion rate in the corona, as shown in Figure 4, is 10⁻⁴ to 10⁻³ in the major radiation region, which produces viscous heat of 10⁻⁴ to 10⁻³ (in units of Eddington luminosity). Even if all of this viscous heating directly goes to electrons, it is much smaller than the additional heating 0.02. The ratio is in the range of 0.5% to 5%. This ratio is nearly the same for a higher f since the accretion rate in the corona increases with increasing f. For an accretion rate higher than 0.2, the ratio is smaller, which can be derived from the accretion rate of the corona shown in the right panel of Figure 4. Therefore, the neglect of viscous heating directly to electrons is a reasonable approximation.

When the accretion is greater than $0.1\,\dot{M}_{\rm Edd}$, the line-driven wind and radiation-pressure-driven wind will be important, which results in the accretion rate decreasing with the decrease in radius as $\dot{M}\propto R^{a}$ (0.0 < a < 1.0) (see Proga et al. 2000; Ohsuga et al. 2005). It is clear that with wind loss, the soft-photon field becomes weaker than the case without winds. Therefore, the Compton cooling is weaker, leading to less condensation. When the system reaches equilibrium, the spectrum can be harder. This is the case without additional energy input. In this work, additional energy is added into the corona and we find that the larger the f is, the harder the spectrum is. Since 0 < a < 1, the energy generating rate ${\propto} ({{\it GM} \dot{M}}/{R}) \propto ({\dot{M}_{\rm out}}/{R^{1-a}})$ is still dominated by the innermost region. Thus, the wind loss causes a decrease of disk luminosity and hence a harder spectrum. However, if wind mass loss is very significant, then there could no longer be an optically thick disk and the effects are large.

A corona lying above a cool disk can illuminate the disk, which increases the energy density of soft photons, leading to stronger coronal radiation. This effect can be significant in the case of a strong corona above a truncated disk. However, for a relatively weak corona above a strong disk, as in the case of HLAGNs, the effect of irradiation is unimportant. We estimate this effect as follows.

Photons emitted from corona are roughly isotropic, about half of which go down to a slab-like disk, irradiating the disk. The irradiating flux is

$$\begin{equation} F_{\rm irr}=\frac{1}{2}F_{{\rm c}}(1-a)=\frac{1}{2}f F_{\rm tot}(1-a), \end{equation} \tag{ 31 }$$

where a is the albedo, which is usually taken as 0.15 (e.g., Zdziarski et al. 1999). This flux is scattered/absorbed in the disk and eventually re-emitted, contributing to the soft photon field. The ratio of this illuminating flux to the disk radiation, F_d = F_tot(1 − f), is

$$\begin{equation} {F_{\rm irr}\over F_{\rm d}}={\frac{1}{2}f(1-a)\over (1-f)}. \end{equation} \tag{ 32 }$$

This ratio reaches 1 when f > 0.7, indicating that the irradiation becomes dominant only when a large fraction of accretion energy is released in the corona. Therefore, we do not include the irradiation in this work in order to make the numerical calculations simple.

Hirose et al. (2006) performed the shearing box numerical simulation with radiative transport, focusing on the vertical structure of the standard thin disk around 300R_g (R_g = (GM/c²)). They found that there is actually very little dissipation in the coronal region as the corona is magnetically dominated. Consequently, the corona is too weak to explain the hard-X-ray emission observed in AGNs. The disk corona evaporation/condensation model without additional heating gives similar results to an MHD simulation (for details see Meyer-Hofmeister et al. 2012). When a corona is heated only by viscous dissipation within the corona, efficient inverse Compton scattering of the disk photons leads to overcooling of the corona. As a consequence, part of the coronal gas condenses into the disk, leaving a very weak corona.

To prevent the corona from collapsing, we assume in this paper that a fraction of accretion energy is transported from the disk and released into the corona, which conflicts with the MHD simulation results. We note that in the MHD simulation of Hirose et al. (2006), it was presumed that the disk is dominated by gas pressure, which is reasonable for studying the vertical structure of black hole X-ray binaries in about 300R_g. However, we are investigating the inner region of an AGN disk, where the disk is radiation-pressure dominant. Recently, Blaes et al. (2011) simulated such a kind of accretion flow and found that the thermodynamics of a radiation-pressure dominant disk differ significantly from those envisaged in standard static models of accretion disks. In radiation-dominant plasma, the buoyant motions become significant for the overall energetics of the plasma. The photons' outward advection becomes comparable to radiative diffusion, and the associated vertical expansion work balances vertical heat transport and dissipation. At the present stage, it is not clear how much discrepancy exists between the MHD simulation and our results.

Based on the thermal stability analysis presented in Yuan (2003), luminous hot accretion flows (LHAFs; Yuan 2001) could also have a two-phase structure. Yuan & Zdziarski (2004) calculated the emitted luminosity from such a kind of two-phase accretion flow and found that it can produce X-ray luminosity as high as 10% Eddington luminosity. This is an alternative model for AGNs with intermediate spectrum and Eddington ratio. In our work, we aim at a more general case for HLAGNs with luminosity up to Eddington value, and calculate the broad waveband spectra from both the disk and the corona.