A general relativistic model of accretion disks with coronae surrounding Kerr black holes

# A general relativistic model of accretion disks with coronae surrounding Kerr black holes

## Abstract

We calculate the structure of a standard accretion disk with corona surrounding a massive Kerr black hole in general relativistic frame, in which the corona is assumed to be heated by the reconnection of the strongly buoyant magnetic fields generated in the cold accretion disk. The emergent spectra of the accretion disk-corona systems are calculated by using the relativistic ray-tracing method. We propose a new method to calculate the emergent Comptonized spectra from the coronae. The spectra of the disk-corona systems with a modified -magnetic stress show that both the hard X-ray spectral index and the hard X-ray bolometric correction factor increase with the dimensionless mass accretion rate, which are qualitatively consistent with the observations of active galactic nuclei (AGNs). The fraction of the power dissipated in the corona decreases with increasing black hole spin parameter , which leads to lower electron temperatures of the coronas for rapidly spinning black holes. The X-ray emission from the coronas surrounding rapidly spinning black holes becomes weak and soft. The ratio of the X-ray luminosity to the optical/UV luminosity increases with the viewing angle, while the spectral shape in the X-ray band is insensitive with the viewing angle. We find that the spectral index in the infrared waveband depends on the mass accretion rate and the black hole spin , which deviates from expected by the standard thin disk model.

accretion, accretion disks, black hole physics, magnetic fields, galaxies: active
4

## 1 Introduction

Active galactic nuclei (AGNs) are powered by the accretion of gases on to their central massive black holes, and the emission of the accretion disks is responsible for the observed multi-band spectral energy distributions (SEDs). It is suggested that the optically thick and geometrically thin accretion disks are present in luminous AGNs, and the observed UV/optical continuum emission of AGNs is the thermal emission from the standard thin accretion disks (e.g., Shakura & Sunyaev, 1973; Shields, 1978; Malkan & Sargent, 1982; Sun & Malkan, 1989; Laor & Netzer, 1989; Laor, 1990; Czerny et al., 2011; Hu & Zhang, 2012). One feature in the spectra of such standard thin accretion disks surrounding massive black holes is, , in the infrared waveband (Koratkar & Blaes, 1999). As the accretion disk extends to the region very close to the black hole, the general relativistic thin accretion disk models were developed in the previous works (e.g., Novikov & Thorne, 1973; Page & Thorne, 1974; Laor & Netzer, 1989; Li et al., 2005; Abramowicz & Fragile, 2011), which were applied successfully to fit the observed SEDs of AGNs and the values of black hole spin parameter can be derived for some sources (e.g., Laor, 1990; Czerny et al., 2011). The values of the spin parameter were estimated by the comparison of the theoretical accretion disk spectra with the observed spectra in some X-ray binaries (Zhang et al., 1997; Steiner et al., 2009; Gou et al., 2011; Kulkarni et al., 2011; McClintock et al., 2011). However, the temperature of the standard thin accretion disks is too low to produce the observed power-law spectra of AGNs in the hard X-ray waveband, and the accretion disk-corona model was therefore suggested (e.g., Galeev et al., 1979; Haardt & Maraschi, 1991; Svensson & Zdziarski, 1994). In this scenario, the power-law hard X-ray spectra of AGNs are most likely due to the inverse Compton scattering of soft photons on a population of hot electrons in the corona above the disk. The soft photons are emitted from the cold disk, and most of them pass through the optically thin corona without being scattered, which emerge as the observed optical/UV continuum emission of AGNs. A small fraction of the photons from the cold disk are Compton scattered by the hot electrons in the corona to the hard X-ray energy band. The disk-corona model was extensively explored in many previous works (e.g., Haardt & Maraschi, 1991; Svensson & Zdziarski, 1994; Kawaguchi et al., 2001; Liu et al., 2002; Cao, 2009). Most gravitational energy is generated in the cold disk through the turbulence, which is probably produced by the magnetorotational instability (Balbus & Hawley, 1991). A fraction of the energy generated in the cold disk should be transported to the corona. One of the most promising mechanisms is the energy of the strongly buoyant magnetic fields in the disk being transported vertically to heat the corona above the disk with the reconnection of the fields (e.g. Di Matteo, 1998; Di Matteo et al., 1999; Merloni & Fabian, 2001). The temperature of the hot electrons in the corona is roughly around 10 K, which can successfully reproduce a power-law hard X-ray spectrum as observed in AGNs (e.g. Liu et al., 2003; Cao, 2009).

The emergent spectrum of an accretion disk surrounding a rotating black hole is influenced by the relativistic effects due to the strong gravity field of the hole, which was extensively investigated in many works (e.g., Zhang et al., 1985; Laor & Netzer, 1989; Laor, 1991; Li et al., 2005, 2009). The iron fluorescence lines observed in some AGNs are significantly asymmetric, characterized with a steep blue wing, an extended red wing and the blueshifted emission line peak (see Miller, 2007, for a review, and the references therein). These features can be modeled as emission from the inner region of an accretion disk illuminated by the external X-ray radiation with the Doppler boost effect and the general relativistic effects (frame-dragging, gravitational redshift and bending of the light) being properly considered (e.g., Laor, 1991). Such effects should also be considered in the calculations of the emergent continuum spectrum of an accretion disk surrounding a rotating black hole observed at infinity (e.g., Laor & Netzer, 1989; Zhang et al., 1997; Laor, 1991; Li et al., 2005, 2009). The ray-tracing method is widely adopted to derive the photon trajectories bent by the strong gravity field of the black hole in the calculations of the emergent spectra emitted from the disk (e.g., Chandrasekhar, 1983; Rauch & Blandford, 1994; Cadez et al., 1998; Fuerst & Wu, 2004; Yuan et al., 2009; Abramowicz & Fragile, 2011). The values of black hole spin parameter can be estimated by the comparison of the observed spectra with the theoretical model calculations (e.g., Yuan et al., 2010). Besides the radiation from the disk, the emergent spectrum of the corona is also affected by these general relativistic effects, which is more complicated than the disk case due to the complexity of the radiation transfer of Comptonized photons in the geometrically thick corona (e.g., Schnittman & Krolik, 2010).

In this work, we calculate the structure of a thin accretion disk with corona surrounding a massive Kerr black hole in general relativistic frame, in which the corona is assumed to be heated by the reconnection of the magnetic fields generated by buoyancy instability in the cold accretion disk. The emergent spectra of the accretion disk-corona systems are calculated by using the relativistic ray-tracing method. The emergent Comptonized spectra from the coronae are calculated by dividing layers in the coronae. We summarize the calculations of the disk-corona structure and its emergent spectrum in Sections 2-4. The numerical results of the model calculations and the discussion are given in Sections 5 and 6.

## 2 The accretion disk-corona structure

In the Boyer-Lindquist coordinates with natural units , the Kerr metric is

 ds2=−e2νdt2+e2ψ(dϕ−ωdt)2+e2μ1dr2+e2μ2dθ2 (1)

where

 e2ν=ΣΔξ,e2ψ=ξsin2θΣ,e2μ1=ΣΔ,e2μ2=Σ,ω=2Mbharξ,
 Δ=r2−2Mbhr+a2,Σ=r2+a2cos2θ,ξ=(r2+a2)2−a2Δsin2θ, (2)

is the mass of black hole, and is the black hole spin parameter . The event horizon of the black hole is

 r+=Mbh+(M2bh−a2)1/2. (3)

The Keplerian velocity of the matter moving around the black hole in circular orbits in the equatorial plane at radius is given by

 ΩK=±M1/2bhR3/2±aM1/2bh, (4)

where the upper sign refers to direct orbits, i.e., corotating with the black hole spinning direction, and the lower sign refers to retrograde orbits, i.e., counterrotating with the black hole spinning direction. The innermost of a geometrically thin accretion disk is assumed to extend to the marginally stable circular orbit of the black hole, which is given by

 rms=Mbh{3+Z2∓[(3−Z1)(3+Z1+2Z2)]1/2}, (5)

where

 Z1=1+(1−a2M2bh)1/3[(1+aMbh)1/3+(1−aMbh)1/3],

and

 Z2=(3a2M2bh+Z21)1/2.

In this work, we consider the relativistic accretion disk with corona, in which the cold disk is optically thick and geometrically thin. The circular motion of the matter in the accretion disk is assumed to be Keplerian, as that in the standard thin disk model (Shakura & Sunyaev, 1973). The formulation of the relativistic accretion disk model is similar to the standard thin disk model, in which some general relativistic correction factors , , , , and , are included (Novikov & Thorne, 1973; Page & Thorne, 1974). The values of all these quantities approach unity in the region far from the black hole (see Abramowicz & Fragile, 2011, for a review).

The gravitational power of the matter dissipated in unit surface area of the accretion disk is

 Q+dissi=3GMbh˙M8πR3LBC1/2, (6)

where is the mass accretion rate of the disk. A fraction of the dissipated power is transported to the corona most probably by the magnetic fields generated in the disk. The fields are strongly buoyant, and the corona above the disk is heated with the reconnection of the magnetic fields (Di Matteo, 1998). The soft photons emitted from the disk are Compton scattered by the hot electrons in the corona to high energy, which is the main cooling process in the corona. About half of the scattered photons are intercepted by the disk, part of which are reflected and the remainder are re-radiated as blackbody radiation (Zdziarski et al., 1999). Thus, the energy equation for the cold disk is

 Q+dissi−Q+cor+12(1−f)Q+cor=4σT4d3τ (7)

where is the interior temperature in the mid-plane of the disk, and is the optical depth in the vertical direction of the disk, and the reflection albedo is adopted in all our calculations. The power transported from the disk to the corona is (Di Matteo, 1998)

 Q+cor=pmυp=B28πυp (8)

where is the magnetic pressure in the disk, and is the vertically rising speed of the magnetic fields in the accretion disk. The rising speed is assumed to be proportional to internal Alfven velocity, i.e., , in which is of the order of unity for extremely evacuated magnetic loops. The strength of the magnetic fields in the disk is crucial in our present investigation, which determines the ratio of power dissipated in the corona to that in the disk. This ratio can be constrained by the observed ratio of the X-ray to bolometric luminosities, which seems to support that the values of should not deviate significantly from unity in most AGNs, though a low value of (e.g. ) cannot be ruled out in few individual AGNs with extremely low (see Vasudevan & Fabian, 2007; Cao, 2009). The detailed physics for generating magnetic fields in the accretion disks is still quite unclear, and we assume the magnetic stress tensor to be related to the pressure of the disk, as done in many previous works (e.g., Sakimoto & Coroniti, 1981; Stella & Rosner, 1984; Taam & Lin, 1984; Hirose et al., 2009). Cao (2009) constructed accretion disk-corona models with different magnetic stress tensors, and found that the model calculations can qualitatively reproduce the observational features that both the hard X-ray spectral index and the hard X-ray bolometric correction factor increase with the Eddington ratio, if ( is adopted). We use this magnetic stress tensor in most of our calculations in this work.

The angular momentum equation for the gas in the disk is (Novikov & Thorne, 1973)

 4πHdτrφ=˙M√GMbhR3FD, (9)

where is the scale height of the accretion disk and the factor is first introduced in Riffert & Herold (1995) in the form of a function . In the vertical direction of the accretion disk, the hydrostatic equilibrium is assumed. The vertical pressure gradient being balanced with the vertical component of the gravity leads to (Riffert & Herold, 1995; McClintock et al., 2006)

 dpdz=−ρGMbhzR3JC (10)

which is valid for the geometrically thin disk, and . Thus the scale height of the disk is given by

 Hd=cs(R3GMbhCJ)1/2, (11)

where the sound speed .

The continuity equation of the disk is

 ˙M=−4πRHdρVrD1/2, (12)

where is the mean density of the disk, is the radial velocity of the accretion flow at radius , and the mass accreted in the corona is neglected.

The state equation of the gas in the disk is

where is the radiation constant, the mean atomic mass . We assume the plasma to consist of 3/4 hydrogen and 1/4 helium, i.e., and .

The ions and electrons in the corona are heated by the reconnection of the magnetic fields rising from the disk, of which the cooling is dominated by the inverse Compton scattering of the soft photons from the disk, the synchrotron radiation, and the bremsstrahlung radiation. The energy equation of the two-temperature corona is given by (Di Matteo, 1998)

 Q+cor=Qiecor+δQ+cor=F−cor, (14)

where is the cooling rate in unit surface area of the corona, and is the energy transfer rate from the ions to the electrons in the corona via Coulomb collisions, which is given by Stepney & Guilbert (1983). We adopt the fraction of the energy directly heating the electrons in all our calculations (Bisnovatyi-Kogan & Lovelace, 1997, 2000). The cooling terms and are the functions of the magnetic field strength, the number density and temperature of the electrons in the corona, which are taken from Narayan & Yi (1995). The cooling rate of the Compton scattering in the corona is calculated by using the approach suggested by Coppi & Blandford (1990).

The state equation of the gas in the optically thin corona is

 pcor=ρcorkTiμimp+ρcorkTeμemp+pcor,m, (15)

where and are the temperatures of the ions and electrons in the two-temperature corona, and the magnetic pressure . Given uncertainty of the magnetic fields in the corona, we simply assume the magnetic pressure to be equipartition with the gas pressure in the corona (Cao, 2009). The cooling of the corona is always dominated by the inverse Compton scattering, and the strength of the fields in the corona mainly affects the synchrotron radiation. Our final results are almost independent of the value of the field strength adopted, except for the spectra in radio wavebands. The scale height of the corona can be estimated based on the assumption of static hydrodynamical equilibrium in the vertical direction.

As done in the work of Cao (2009), we need to specify the ion temperature of the corona in the calculations of the disk-corona structure. It is found that the ion temperature has little influence on the X-ray spectra, and Liu et al. (2003) pointed out that the temperature of the ions in the corona is always in the range of 0.2-0.3 virial temperature . In this work, the definition of the virial temperature is slightly different,

 32kTvir=12mpΩ2kR2, (16)

where is the Keplerian velocity of the gas in the disk at radius . In this work, we adopt in all our calculations, which is almost equivalent to the value suggested by Liu et al. (2003).

It is still quite unclear on the outer radius of the accretion disk. One conventional estimate of the outer radius of the disk is based on the assumption that the disk is truncated where it becomes gravitational unstable (e.g., Collin & Zahn, 2008). The outer radius of the disk is estimated with the Toomre parameter(Toomre, 1964; Goldreich & Lynden-Bell, 1965),

 QToomre=ΩKcs2πGρHd=1. (17)

Given the values of the black hole mass , the spin parameter , the viscosity parameter , and the mass accretion rate , the disk structure can be derived by solving Equations (6)-(8) numerically. The structure of the corona is then available based on the derived disk structure by solving Equations (14)-(16).

## 3 Ray tracing method

The trajectory of the photons emitted from the inner region of the accretion disk-corona system is bent by the strong gravity of the black hole. The observed emergent spectrum of such an accretion disk-corona system surrounding a Kerr black hole can be calculated with the ray tracing method. We summarize the method briefly in this section.

In the Kerr spacetime, there are four constants for the motion of a photon (Carter, 1968; Abramowicz & Fragile, 2011), three of which can determine the trajectory of the photon, namely the energy at infinity,

 E=−pt, (18)

the effective angular momentum,

 Lz=pϕ, (19)

and the Carter constant,

 Q=p2θ−a2E2cos2θ+L2zcot2θ, (20)

where is the four-momentum of the photon. Combining Equations (18)-(20) and the expressions of a photon’s four-momentum in Carter (1968), we derive the following equations describing the trajectory of a photon in the Kerr spacetime:

 Σdrdσ = ±√R(r), (21) Σdθdσ = ±√Θ(θ), (22) Σdϕdσ = Lzsin2θ+2arE−Lza2Δ, (23) Σdtdσ = −a(aEsin2θ−Lz)+(r2+a2)[E(r2+a2)−Lza]Δ, (24)

where

 R(r) = [E(r2+a2)−Lza]2−Δ[(Lz−aE)2+Q], (25) Θ(θ) = Q−(−a2E2+L2zsin2θ)cos2θ. (26)

Here is the affine parameter along the trajectory. The signs in the above equations must be the same as the signs of , respectively, and they change at the turning point in or , i.e., or . As the trajectory of a photon is independent of its energy , Cunningham & Bardeen (1973) introduce two dimensionless parameters being conserved along the trajectory,

 λ≡LzE,q≡QE2 (27)

and define another affine parameter . The equations describing the trajectory of the photon can be re-written as

 dr = ±√˜R(r)dP, (28) dθ = ±√˜Θ(θ)dP, (29) dϕ = [λsin2θ+2ar−λa2Δ]dP, (30) dt = [(r2+a2)2Δ−a2sin2θ−2arλΔ]dP, (31)

where , . Integrating the null geodesic equations of the photon along the path from the initial location to the observer at infinity with position , the trajectory can be determined by the following integral equations

 ±robs∫remdr√˜R(r)=±θobs∫θemdθ√˜Θ(θ)=P, (32)

and

 ϕobs−ϕem=±Pobs∫Pem(λsin2θ+2ar−a2λΔ)dP, (33)

where , are functions of or , corresponding to the upper limit value of the -integral taken as , respectively, and the signs in the above integrals must be the same as those of and to guarantee the integral is always positive and increasing along a photon’s trajectory. In the calculations, we set , and . The detailed explanation on the integrals can be found in some previous works (Chandrasekhar, 1983; Cadez et al., 1998; Li et al., 2005).

The image of the accretion disk-corona observed at an angle with respect to the axis of the black hole at infinity can be described by the two impact parameters,

 α=−(rp(ϕ)p(t))r→∞=−λsinθobs, (34)

and

 β=(rp(θ)p(t))r→∞=(q+a2cos2θobs−λ2cot2θobs)1/2=pobs, (35)

where is the apparent displacement of the image perpendicular to the projected axis of the black hole, and is the apparent parallel displacement. The vector is the four-momentum in the local nonrotating frame(LNRF). The observed image of the accretion disk-corona can be calculated with the ray tracing method described above, provided that the local spectrum of the disk-corona is known. For a given point (, ) in the image seen by the observer at (), the two constants of motion and can be derived with Equations (34) and (35). Then the trajectory integral , the radius , and the azimuthal angle , can be calculated by using Equations (32) and (33).

## 4 Radiation transfer and emergent spectrum

In this work, the photons cannot move through the equatorial plane because of the optically thick accretion disk, and therefore we neglect such trajectories. For the photons passing through the corona, the optical depth for the scattering of photons in the corona in the curved space can be calculated with the ray tracing method. Along the trajectory of the photon in the corona, the differential optical depth is

 dτ=σTnedl, (36)

where is the Thompson cross-section, is the electrons number density in the corona at , and

 dl=eνΣdP=±eνΣ√~Θ(θ)dθ. (37)

Integrating along the photon trajectory of the photon in the corona, we have

 τ=±θobs∫θemσTneeνΣ√~Θ(θ)dθ, (38)

where .

As the accretion disk is geometrically thin, we approximate in calculating the emergent spectrum of the disk. The radiative flux from the accretion disk observed at infinity is given by

 F(νobs)=∫e−τBνobsdΩ=∫e−τBνobsdαdβD2, (39)

where is the intensity at observed frequency , is the solid angle subtended by the image of the disk.

The gravitational redshift effect and Doppler redshift can be included by using the Lorentz invariant (Rybicki & Lightman, 1979), so the emergent spectrum can be calculated with

 F(νobs)=∫g3e−τBνedαdβD2 (40)

where is the the intensity of the photons from the disk at emitting frequency . Integrating over the disk image, the emergent spectrum of the disk is available with

 Ldisk(νobs)=4πD2F(νobs)=4π∫g3e−τBνedαdβ. (41)

The redshift factor is defined by (Cunningham & Bardeen, 1973)

 g≡νobsνe=pμuμ|obspμuμ|em, (42)

where is the four-momentum of the photon (Carter, 1968),

 pμ=(pt,pr,pθ,pϕ)=(−1,±√˜R(r)Δ,±√˜Θ(θ),λ)E. (43)

The four-velocity of the fluid in the locally non-rotating frame (LNRF) and the local rest frame (LRF) are given by

 uμobs=(1,0,0,0), (44)

and

 uμem=(γrγϕe−ν,γrβre−μ1,0,γrγϕ(ωe−ν+βϕe−ψ)), (45)

respectively (Yuan et al., 2009).

The corona is assumed to rotate with the disk at Keplerian velocity. The calculations of the emergent spectrum of the corona is slightly different from that of the accretion disk, because the redshift factor and the trajectory of a photon depends on the location of the photon emitted in the corona. To calculate the redshift factor and the trajectory of such a photon, we numerically solve Equations (32) and (33) to derive the impact parameters and . We divide the corona into small volumes with where is element area and is the thickness of a layer, and the emergent spectrum of each volume is available by calculating the radiation transfer with the ray tracing method, if the emissivity of the corona is available. For simplicity, we assume the radiation of the corona to be isotropic and the emissivity is homogeneous in the vertical direction of the corona.

The synchrotron and bremsstrahlung emissivities are taken from Narayan & Yi (1995). The spectrum of synchrotron and bremsstrahlung emission from the corona is available by summing up the contribution from these small volumes,

 Lsyn,brem=4π∫g3he−τh(εsynνe+εbremνe)dαdβ, (46)

where the effects of the absorption and the scattering in the corona are considered as described at the beginning of this section.

In the calculations of the Comptonization in the corona, we assume the corona to be a parallel plane (see Cao, 2009, for the details). The mean probability of the soft photons injected from the cold disk experiencing first-order scattering in the corona is

 P1=21∫0(1−e−τ0/cosθ)cosθdcosθ, (47)

where the constant specific intensity of the soft photons from the disk is assumed, is the angle of the motion of the soft photons with respect to the vertical direction of the disk, and is the optical depth of the corona for electron scattering in the vertical direction. We can calculate the first-order Comptonized spectra of the corona using the method suggested by Coppi & Blandford (1990) with the probability given by Equation (47), if the density and the temperature of electrons in the corona, and the incident spectrum of the cold disk are known. As we have assumed the scattered photons to be isotropic, the emissivity of the first-order Compton scattered photons in the layer with the height between and can be approximated as . The first-order Compton scattered photons will pass though the corona before arriving the observer. Using the above described ray tracing approach, we can calculate the emergent first-order Comptonized spectrum of the corona.

The mean probability for these first-order scattered photons experiencing the second-order scattering can be estimated as

 P2=121∫0dξ1∫0[1−e−ξτ0/cosθ+1−e−(1−ξ)τ0/cosθ]dcosθ, (48)

where , and the first-order scattered photons are assumed to be radiated isotropically. The two terms in the integration are for downward-moving and upward-moving photons emitted at height , respectively. The probability of the scattered photons experiencing next higher order scattering approximates to , so we simply adopt for . Thus, the th-order Comptonized spectra and emissivity can be derived with the same method for calculating the first-order Comptonization, in which the th-order Comptonized spectra are used instead of the incident spectrum from the cold disk used for the calculation of the first-order Comptonization. In the same way, the emergent spectra of high order Comptonized spectra are derived, and the total Comptonized spectrum of the corona is available by summing up the contribution from the whole corona,

 LComp=4π∫g3h(εCompν,1+εCompν,2+...+εCompν,n)dαdβ. (49)

## 5 Results

As described in Section 2, we can calculate the structure of an accretion disk with corona surrounding a rotating Kerr black hole, if the black hole mass , the spin parameter , and the dimensionless accretion rate are specified. Based on the derived structure of the accretion disk-corona system, the emergent spectrum observed at infinity with an inclination angle is calculated with the ray tracing method described in Sections 3 and 4, in which all the general relativistic effects are considered. The black hole mass is adopted in most of our calculations for AGNs. The dimensionless mass accretion rate is defined as () independent of black hole spin.

We plot the ratios of the power dissipated in the corona to the total released gravitational power as functions of the accretion rate with different values of the black hole spin parameter in Figure 1. It is found that the dependence of the ratio with is quite different for the cases with different magnetic stress tensors, which are qualitatively consistent with those obtained in Cao (2009) for the Newtonian accretion disk corona model. We find that the ratio decreases with increasing spin parameter with either or if all other parameters are fixed, which means that the efficiency of the energy transportation from the disk to the corona is low for a rapidly spinning black hole independent of the detailed magnetic stress tenor adopted. However the model with appears to show the opposite trend that reaches large values for a rapidly spinning black hole.

In Figure 2, we plot the interior temperatures of the accretion disks as functions of radius with different values of the accretion rate and black hole spin parameter , and the stress is adopted in the calculations. It is not surprising that the temperature of the disk increases with the mass accretion rate . For a rapidly spinning black hole, the inner edge of the disk extends to the region close to the black hole, and the temperature of the gas in the inner edge of the disk is significantly higher than that for a slowly spinning (or non-spinning) black hole.

The structures of the corona calculated with , i.e., the temperatures of the ions and electrons, and the electron scattering optically depth of the corona in the vertical direction, are plotted in Figure 3. The electron temperature decreases with increasing mass accretion rate and the black hole spin parameter . The optical depth for electron scattering is in the range of for different values of and adopted.

The emergent spectra of the accretion disk-corona systems with observed at infinity with different disk model parameters and viewing angles are plotted in Figure 4. In each panel, one of these parameters is fixed at a certain value and the other ones are taken as free parameters in the calculations of the emergent spectra. The corona is optically thin and geometrically thick, and therefore its radiation is almost isotropic. The radiation of the disk is dominantly in the optical/UV bands, the emergent spectrum of which depends on the viewing angle sensitively due to its slab-like geometry. The photon spectral indices, and the ratios of the bolometric luminosity to the X-ray luminosity in  keV as functions of the accretion rate for different values of and are plotted in Figure 5. The hard X-ray spectral index increases with the accretion rate and black hole spin , while it is insensitive with inclination . The ratio of the bolometric luminosity to the X-ray luminosity increases with mass accretion rate, which is higher for a rapidly spinning black hole.

The calculated emergent spectrum is usually insensitive to the outer radius of the accretion disk, except the spectral shape in the infrared/optical wavebands. The outer radius of the accretion disk can be estimated by assuming the disk to be truncated where the self-gravity of the disk dominates over the central gravity of the black hole. The outer radii of the accretion disks with different accretion rates and black hole masses are plotted in Figure 6. The results show that the accretion disk surround a less massive black hole extends to a larger radius. The spectral shape of the accretion disk in the infrared/optical wavebands also varies with the outer radius of the disk . We plot the emergent spectra of the disk-corona systems with in the near-infrared/optical wavebands with different values of the accretion rate and black hole spin parameter in Figure 7. We find that the spectral shapes of the accretion disk-corona systems deviate significantly from , which is predicted by the standard accretion disk model (Shakura & Sunyaev, 1973; Koratkar & Blaes, 1999; Hubeny et al., 2000). Davis et al. (2007) measured UV spectral slopes (, where ) for several thousand quasars from the Sloan Digital Sky Survey (SDSS), and the distributions of the slope span a wide range rather than predicted by the standard accretion model.

## 6 Discussion

In this work, we calculate the structure of an accretion disk with corona around a Kerr black hole, and the emergent spectrum of the system is derived with the ray tracing approach, in which the general relativistic effects have been considered. The ions and electrons in the corona are assumed to be heated by magnetic power with the reconnection of the buoyant magnetic fields transported from the disk (Di Matteo, 1998). The strength of the magnetic fields can be estimated from gas/radiation pressure of the disk with assumed magnetic stress tensor. We examine the relations of the ratio with the mass accretion rate in the models with different magnetic stress tensors (see Figure 1). In order to understand the results, we analyze the situation based on the standard thin disk model. The gravitational power dissipated in unit surface area of the accretion disk is given by (Shakura & Sunyaev, 1973)

 Q+dissi=12WrφRdΩkdr=32τrφHdΩk=32τrφcs=32τrφ(ptot+pmρ)1/2 (50)

where , . The power dissipated in the corona is

 Q+cor=pmυp=bυApm=bpmB(4πρ)1/2=bpm(2pmρ)1/2 (51)

Combining Equations (50) and (51), we find that the ratio . If the stress tensor is adopted, the ratio remains constant independent of accretion rate . For the cases of , or , the ratio or respectively, as , and the radiation pressure is dominant in the inner regions of the accretion disk. As the disk temperature increases with mass accretion rate , the ratio therefore decreases with increasing . The results obtained in this work for spinning black holes are qualitatively consistent with those derived in Cao (2009).

As the value of black hole spin parameter increases, the disk extends close to the black hole, and more gravitational power is released in the inner region of the accretion disk. The temperature of the inner disk region increases with the spin parameter , which increases the energy density of the soft photons emitted from the disk. This leads to the corona cooling efficiently, because the cooling of the corona is dominated by the Compton scattering of the soft photons from the disk. The electron temperature of the corona in the inner region of the disk is therefore lower for rapidly spinning black holes (see Figure 3).

The resulted spectra of the disk-corona systems with magnetic stress tensor show that both the hard X-ray spectral index and the hard X-ray bolometric correction factor increase with the Eddington ratio, which are qualitatively consistent with the observations of AGNs (Wang et al., 2004; Shemmer et al., 2006; Vasudevan & Fabian, 2007), and X-ray binaries (Wu & Gu, 2008). The observed ratio of may provide additional information of the disk-corona systems (Davis & Laor, 2011). The stress tensor was initially suggested by Taam & Lin (1984) based on the idea that the viscosity is proportional to the gas pressure, but the size of turbulence should be limited by the disk scale height determined by the total pressure. This is also supported by the analysis on the local dynamical instabilities in magnetized, radiation-pressure-supported accretion disks (Blaes & Socrates, 2001; Merloni & Fabian, 2002). However, recent radiative MHD simulations seem to prefer the stress tensor (Hirose et al., 2009). Our model calculations can be carried out when the form of magnetic stress tenor is specified. For simplicity, in most cases of this work, we calculate the structure and emergent spectra of accretion disk-corona systems using the magnetic stress tensor .

The radiation from the corona is almost isotropic due to its optically thin and geometrically thick structure, while the radiation from the disk is anisotropic. It is therefore found that the hard X-ray spectral index is insensitive with the viewing angle . The X-ray spectra are mainly dominated by the inverse Compton scattering, which mainly depends on the temperature of the electrons. Therefore the trend of the photon spectral indices and the X-ray correction factors can be explained by the decrease of the temperature of the electrons with increasing mass accretion rate , or the black hole spin parameter (see Figure 5, and also Figure 3). The photon index of the hard X-ray spectrum becomes large when the black hole is rapidly spinning if all other parameters are fixed, which is due to the temperature of the electrons decreases with increasing value of . The correction factor derived by the model calculations with are significantly larger than the observed values when (e.g., Wang et al., 2004; Vasudevan & Fabian, 2007). One possibility is that an advection dominated accretion flow (ADAF) is present in the inner region of the disk (Narayan & Yi, 1994, 1995). The inner ADAF is X-ray luminous, which leads to a low value of (Quataert et al., 1999). This is also consistent with the fact that the X-ray spectra of the AGNs are hard when accreting at low rates (see Wang et al., 2004; Cao, 2009, for the detailed discussion). An alternative resolution is the model with magnetic stress tenor , however, it always predicts a constant independent of mass accretion rate . This needs further investigations in the future.

In Figure 7, the spectral shapes of the disk-corona model with different values of accretion rates and black hole spin parameter in the near-infrared/optical bands are plotted, which is dominantly contributed by the blackbody radiation from the disk. The spectral shapes are significantly different from , as expected in standard thin disk model. This is mainly caused by the -dependent temperature distributions in the disks surrounding massive black holes and the outer radii of the disks being different from those in standard thin disk model. In the standard thin disk model, the spectral shape in the near-infrared/optical wavebands is derived by assuming the temperature of the accretion disk and the disk to extend to infinity. However, the temperature profiles of relativistic accretion disks deviate significantly from in the inner region of the disk () (see Figure 2). The contribution to the spectra in IR/optical bands from the blackbody radiation of the outer region of the disks can also be quite important. The outer radius of the disk is a function of black hole mass and accretion rate (see Figure 6). The observational slope of the infrared continua can be used to constrain the accretion disk model (see Kishimoto et al., 2008). It is not suitable to compare the observed spectra only with, , as predicted by the standard accretion disk model. In this work, the local radiation from the disk is approximated as blackbody emission. A better approximation is to include a spectral hardening factor in calculating the local spectrum of the disk (e.g., Shimura & Takahara, 1995; Hubeny et al., 2000; Davis et al., 2007), with which the energy peak of the calculated disk spectrum will shift to a slightly higher frequency, while the spectral shape changes little. This would be important for detailed fitting the observed SED of AGNs, which is beyond the scope of this paper.

It is assumed that zero torque at the inner edge of the disk (Shakura & Sunyaev, 1973), which was doubted in some previous works (Agol & Krolik, 2000; Hawley & Krolik, 2001). The general relativistic magnetohydrodynamic (GRMHD) simulations provide an effective tool to explore the torque in the plunging region (Shafee et al., 2008; Kulkarni et al., 2011). In the accretion disk with the thickness the magnetic torque at the radius of the innermost stable circular orbit (ISCO) is only of the inward flux of angular momentum at this radius, which indicates that the zero torque is really a good approximation for geometrically thin disks (Shafee et al., 2008). Gong et al. (2012) investigated the accretion disk-corona model similar to that developed by Cao (2009), in which a non-zero torque at the inner edge of the disk is assumed. As the detailed accretion physics in the plunging region is complicated and still quite unclear, we adopt the assumption of no torque in the inner edge of the accretion disk in all our calculations. The calculations in this work were carried out for massive black holes, which will be compared with the observed spectra of AGNs in our future work. The model calculations can also be applicable to X-ray binaries.

## Acknowledgements

This work is supported by the National Basic Research Program of China (grant 2009CB824800, 2012CB821800), the NSFC (grants grants 11173043, 11121062, 11233006, 11073020, and 11133005), the CAS/SAFEA International Partnership Program for Creative Research Teams (KJCX2-YW-T23), and the Fundamental Research Funds for the Central Universities (WK2030220004).

.

### Footnotes

1. affiliation: Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai, 200030, China; youbei@shao.ac.cn, cxw@shao.ac.cn
2. affiliation: Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai, 200030, China; youbei@shao.ac.cn, cxw@shao.ac.cn
3. affiliation: Department of Astronomy, University of Sciences and Technology of China, Hefei, Anhui 230026, China; yfyuan@ustc.edu.cn
4. slugcomment: re-submitted to ApJ, on August 10, 2019

### References

1. Abramowicz, M. A., & Fragile, P. C. 2011, arXiv:1104.5499
2. Agol, E., & Krolik, J. H. 2000, ApJ, 528, 161
3. Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214
4. Bisnovatyi-Kogan, G. S., & Lovelace, R. V. E. 1997, ApJ, 486, L43
5. Bisnovatyi-Kogan, G. S., & Lovelace, R. V. E. 2000, ApJ, 529, 978
6. Blaes, O., & Socrates, A. 2001, ApJ, 553, 987
7. Cadez, A., Fanton, C., Calvani, M., & Marziani, P. 1998, 19th Texas Symposium on Relativistic Astrophysics and Cosmology,
8. Cao, X. 2009, MNRAS, 394, 207
9. Carter, B. 1968, Physical Review, 174, 1559
10. Chandrasekhar, S. 1983, Research supported by NSF. Oxford/New York, Clarendon Press/Oxford University Press (International Series of Monographs on Physics. Volume 69), 1983, 663 p.,
11. Collin, S., & Zahn, J.-P. 2008, A&A, 477, 419
12. Coppi, P. S., & Blandford, R. D. 1990, MNRAS, 245, 453
13. Cunningham, C. T., & Bardeen, J. M. 1973, ApJ, 183, 237
14. Czerny, B., Hryniewicz, K., Nikołajuk, M., & Sa̧dowski, A. 2011, MNRAS, 415, 2942
15. Davis, S. W., & Laor, A. 2011, ApJ, 728, 98
16. Davis, S. W., Woo, J.-H., & Blaes, O. M. 2007, ApJ, 668, 682
17. Di Matteo, T. 1998, MNRAS, 299, L15
18. Di Matteo, T., Celotti, A., & Fabian, A. C. 1999, MNRAS, 304, 809
19. Fuerst, S. V., & Wu, K. 2004, A&A, 424, 733
20. Galeev, A. A., Rosner, R., & Vaiana, G. S. 1979, ApJ, 229, 318
21. Goldreich, P., & Lynden-Bell, D. 1965, MNRAS, 130, 97
22. Gong, X.-L., Li, L.-X., & Ma, R.-Y. 2012, MNRAS, 420, 1415
23. Gou, L., McClintock, J. E., Reid, M. J., et al. 2011, ApJ, 742, 85
24. Haardt, F., & Maraschi, L. 1991, ApJ, 380, L51
25. Hawley, J. F., & Krolik, J. H. 2001, ApJ, 548, 348
26. Hirose, S., Blaes, O., & Krolik, J. H. 2009, ApJ, 704, 781 2012
27. Hu, R., & Zhang, S.-N. 2012, arXiv:1206.2569
28. Hubeny, I., Agol, E., Blaes, O., & Krolik, J. H. 2000, ApJ, 533, 710
29. Kawaguchi, T., Shimura, T., & Mineshige, S. 2001, ApJ, 546, 966
30. Kishimoto, M., Antonucci, R., Blaes, O., et al. 2008, Nature, 454, 492
31. Koratkar, A., & Blaes, O. 1999, PASP, 111, 1
32. Kulkarni, A. K., Penna, R. F., Shcherbakov, R. V., et al. 2011, MNRAS, 414, 1183
33. Laor, A., & Netzer, H. 1989, MNRAS, 238, 897
34. Laor, A. 1990, MNRAS, 246, 369
35. Laor, A. 1991, ApJ, 376, 90
36. Li, G.-X., Yuan, Y.-F., & Cao, X. 2010, ApJ, 715, 623
37. Li, L.-X., Zimmerman, E. R., Narayan, R., & McClintock, J. E. 2005, ApJS, 157, 335
38. Li, Y.-R., Yuan, Y.-F., Wang, J.-M., Wang, J.-C., & Zhang, S. 2009, ApJ, 699, 513
39. Liu, B. F., Mineshige, S., & Shibata, K. 2002, ApJ, 572, L173
40. Liu, B. F., Mineshige, S., & Ohsuga, K. 2003, ApJ, 587, 571
41. Malkan, M. A., & Sargent, W. L. W. 1982, ApJ, 254, 22
42. McClintock, J. E., Narayan, R., Davis, S. W., et al. 2011, Classical and Quantum Gravity, 28, 114009
43. McClintock, J. E., Shafee, R., Narayan, R., et al. 2006, ApJ, 652, 518
44. Merloni, A., & Fabian, A. C. 2001, MNRAS, 328, 958
45. Merloni, A., & Fabian, A. C. 2002, MNRAS, 332, 165
46. Miller, J. M. 2007, ARA&A, 45, 441
47. Narayan, R., & Yi, I. 1994, ApJ, 428, L13
48. Narayan, R., & Yi, I. 1995, ApJ, 452, 710
49. Novikov, I. D., & Thorne, K. S. 1973, Black Holes (Les Astres Occlus), 343
50. Page, D. N., & Thorne, K. S. 1974, ApJ, 191, 499
51. Quataert, E., Di Matteo, T., Narayan, R., & Ho, L. C. 1999, ApJ, 525, L89
52. Rauch, K. P., & Blandford, R. D. 1994, ApJ, 421, 46
53. Riffert, H., & Herold, H. 1995, ApJ, 450, 508
54. Rybicki, G. B., & Lightman, A. P. 1979, New York, Wiley-Interscience, 1979. 393 p.,
55. Sakimoto, P. J., & Coroniti, F. V. 1981, ApJ, 247, 19
56. Schnittman, J. D., & Krolik, J. H. 2010, ApJ, 712, 908
57. Shafee, R., McKinney, J. C., Narayan, R., et al. 2008, ApJ, 687, L25
58. Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
59. Shemmer, O., Brandt, W. N., Netzer, H., Maiolino, R., & Kaspi, S. 2006, ApJ, 646, L29
60. Shields, G. A. 1978, Nature, 273, 519
61. Shimura, T., & Takahara, F. 1995, ApJ, 445, 780
62. Steiner, J. F., McClintock, J. E., Remillard, R. A., Narayan, R., & Gou, L. 2009, ApJ, 701, L83
63. Stella, L., & Rosner, R. 1984, ApJ, 277, 312
64. Stepney, S., & Guilbert, P. W. 1983, MNRAS, 204, 1269
65. Sun, W.-H., & Malkan, M. A. 1989, ApJ, 346, 68
66. Svensson, R., & Zdziarski, A. A. 1994, ApJ, 436, 599
67. Taam, R. E., & Lin, D. N. C. 1984, ApJ, 287, 761
68. Toomre, A. 1964, ApJ, 139, 1217
69. Vasudevan, R. V., & Fabian, A. C. 2007, MNRAS, 381, 1235
70. Wang, J.-M., Watarai, K.-Y., & Mineshige, S. 2004, ApJ, 607, L107
71. Wu, Q., & Gu, M. 2008, ApJ, 682, 212
72. Yuan, W., Liu, B. F., Zhou, H., & Wang, T. G. 2010, ApJ, 723, 508
73. Yuan, Y.-F., Cao, X., Huang, L., & Shen, Z.-Q. 2009, ApJ, 699, 722
74. Zdziarski, A. A., Lubinski, P., & Smith, D. A. 1999, MNRAS, 303, L11
75. Zhang, J. L., Xiang, S. P., & Lu, J. F. 1985, Ap&SS, 113, 181
76. Zhang, S. N., Cui, W., & Chen, W. 1997, ApJ, 482, L155
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters