COMPTONIZED PHOTON SPECTRA OF SUPERCRITICAL BLACK HOLE ACCRETION FLOWS WITH APPLICATION TO ULTRALUMINOUS X-RAY SOURCES

We solve the axisymmetric two-dimensional radiation hydrodynamic equations in spherical coordinates (r, θ, ϕ) by adopting the flux-limited diffusion approximation (Levermore & Pomraning 1981; Turner & Stone 2001). General relativistic effects are incorporated by a pseudoNewtonian potential (Paczyńsky & Wiita 1980), Ψ = −GM/(r − r_s), where r_s(= 2GM/c²) is the Schwarzschild radius, G is the gravitational constant, M = 10 M_☉ is the mass of the central black hole, and c is the speed of light. We adopt the α prescription of viscosity (Shakura & Sunyaev 1973) and set α = 0.1. Basic equations and numerical methods to solve the radiation hydrodynamic equations are the same as in Kawashima et al. (2009), and they are summarized in Appendix A.

We improve on the investigation by Kawashima et al. (2009) by using a larger computational domain 2r_s ⩽ r ⩽ 1000r_s and 0 ⩽ θ ⩽ π/2, and a higher resolution (N_r, N_θ) = (192, 192). We assume azimuthal symmetry and mirror symmetry with respect to the equatorial plane. The grid points in radial and polar directions are distributed such that Δln r = constant and Δcos θ = 1/N_θ, respectively. We start the calculations with a hot, rarefied, optically thin atmosphere (i.e., without an initial optically thick disk). In order to simulate the formation and the evolution of the accretion disks, we continuously supply mass through the outer boundary at r = 1000r_s near the equatorial plane (0.45π  ⩽  θ  ⩽  0.5π) with a constant rate ${\dot{M}}_{\rm input}$. The injected matter is assumed to have a specific angular momentum corresponding to the Keplerian angular momentum at r = 300r_s. Viscous processes allow the angular momentum of the gas to be transported outward, which drives the gas inflow. Eventually, the gas falls onto the black hole in a quasi-steady fashion. At the outer boundary except for the mass injection region (i.e., r = 1000r_s and 0  ⩽  θ  <  0.45π), we allow matter to escape freely but not to enter the computational domain. We further adopt an absorbing boundary condition at r = 2r_s. We note that the initial hot atmosphere does not affect the results of our calculations because it is swept away from the computational domain by the accretion flow and the outflow.

The left panel of Figure 1 shows the time-averaged distribution of mass density and temperature for a model with accretion rate ${\dot{M}} \,{\approx}\, 200L_{\rm E}/c^{2}$, where L_E is the Eddington luminosity and ${\dot{M}}$ is measured at r = 2r_s. The supercritical accretion flow produces four regions: (1) a radiation pressure dominant accretion disk, (2) a sub-relativistic, mildly hot (T_gas ∼ 10^7.5–10⁸ K) funnel jet around the rotation axis, (3) a cool (T_gas ∼ 10^6.5 K), dense (ρ  ∼ 10⁻⁶ g cm⁻³), and slow (≲ 0.01c) outflow emerging from the disk, and (4) a horn-like hot region around the funnel wall near the black hole (r  ≲  5r_s), where the gas temperature increases to T_gas  ∼  10⁸ K due to shock heating by supersonic inflow bounced by the centrifugal barrier of the potential (the right panel of Figure 1). The reflection shocks are identified by entropy jumps and kinks of the streamlines. Hot regions are also created in the funnel owing to the heating by internal shocks.

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The photon spectra are computed by solving the three-dimensional radiative transfer using the Monte Carlo method (e.g., Pozdnyakov et al. 1983), by post-processing the results of radiation hydrodynamic simulations. Since non-gray radiation hydrodynamic simulations fully coupled with the Monte Carlo radiative transfer require huge computational resources, we calculate photon spectra by post-processing the simulation results. Here, we should emphasize that our radiation hydrodynamic calculations incorporate the effects of Comptonization (i.e., Compton cooling/heating and subsequent change in fluid dynamics) by applying the Kompaneets equation and by solving Equations (A1)–(A6) in Appendix A. Although the Compton cooling/heating rate used in our radiation hydrodynamic simulations might not be fully consistent with the results of full radiative transfer, our spectral calculations do not significantly overestimate the effects of Comptonization on spectra compared to the spectral calculations using the simulation results obtained without taking into account the Compton cooling.

We ignored the curvature of photon trajectories and gravitational redshift because most of the photons affected by these general relativistic effects are trapped by the inflowing plasma, and swallowed by the black hole, as we shall show in Section 3.

We adopt Cartesian coordinates to calculate the radiative transfer in the region −300r_s ⩽ x, y, z  ⩽ 300r_s (depicted by the square box in Figure 1(a)). The grid points are equally spaced with a number of grid points of (N_x, N_y, N_z) = (160, 160, 160). The physical quantities at each grid point are obtained by interpolating the time-averaged simulation results. We averaged the quantities at each grid points for 5 s after accretion flows become quasi-steady. We employed the model for the mass input rate from the outer boundary ${\dot{M}_{\rm input}} =$ 10,000, 3000, and 1000L_E/c², whose eventual mass accretion rates onto the black hole are ${\dot{M}}$  ≈ 1000, 500, and 200L_E/c², respectively.

We calculate radiation spectra by Monte Carlo simulations, incorporating emission, absorption, and the scattering by the following algorithm.

• 1.  
  In order to limit the emission point to avoid too many scatterings in the optically thick region, we identify the surface where the effective optical depth ${\tau }_{\rm eff}={\sqrt{3{\tau _{\rm a}}({\tau }_{\rm a} + {\tau }_{\rm e})}}$ equals 10. Here, τ_a and τ_e are the optical depth for the absorption and the Thomson scattering, respectively, calculated by integrating the opacity in the −z-direction starting from the upper boundary at z = 300r_s.The absorption coefficient is evaluated by free–free absorption coefficient α^ff_ν obtained from Kirchhoff's law ε^ff_ν/4π = α_ν^ffB_ν, where ε^ff_ν is the emissivity of free–free emission (i.e., bremsstrahlung) and B_ν is the intensity of the blackbody radiation at frequency ν. We consider the emissivity of the electron–electron and the electron–proton bremsstrahlung (see Manmoto et al. 1997). For Thomson scattering, we adopted α^es_ν = n_eσ_T, where n_e is the number density of the plasma and σ_T is the cross-section of the Thomson scattering. Since the aim of calculating τ_eff in this calculation step is just restricting the region for emissions of seed photons, for simplicity, we used the cross-section for Thomson scattering and took into account the corrections for α^ff_ν and α^es_ν by fluid motions only (see Appendix B). As we explain later, when we calculate the escape probability in order to trace photon packets, we calculate scattering coefficients by using the Klein–Nishina formula taking into account the correction due to bulk motions and thermal motions of electrons. We carried out calculations limiting the emission region of seed photon packets above the surface τ_eff = 1, 3, 5, 10, 20 and confirmed that the spectral shape of the photons when τ_eff = 10 is essentially the same as that obtained by employing the emission region deeper than τ_eff = 10.It should be noted that the surface where τ_eff = 1 is located inside the disk when the distance from the rotation axis $R= \sqrt{x^{2} + y^{2}}$ satisfies R ≲ (several)× 10r_s, while it is found in the dense outflow in the outer disk. In our calculation, therefore, seed photons created in the inner disk and in the outflow covering the outer disk mainly contribute to the observable spectra.
• 2.  
  We place a seed photon packet above the surface τ_eff = 10. The point where the seed photon is emitted is determined such that the emission probability at each grid cell is proportional to the emissivity, by using the inverse function method for a function defined as

  $$\begin{equation} f(n) = \frac{\sum _{{\ell } {\le } n} {\varepsilon }_{\nu }^{\rm ff}({\ell })}{\sum _{{\ell } {\le } n_{\rm total}} {\varepsilon }_{\nu }^{\rm ff} ({\ell })}, \end{equation} \tag{ 1 }$$

  where n and n_total are the sequential number and the total number of grid points which satisfy the condition τ_eff ⩽ 10, respectively. The emissivity of the seed photon ε^ff_ν is given by the thermal bremsstrahlung emission. The intensity of thermal bremsstrahlung emission approaches that of a blackbody deep inside the disk or in the dense outflow, where τ_eff significantly exceeds 1. We ignored the cyclotron and synchrotron emissivity. This is partly because our radiation hydrodynamic simulation does not include magnetic fields, and partly because the disk or outflow is dense and hot enough to attain blackbody intensity via only the thermal bremsstrahlung emission/absorption in the soft X-ray band (see Section 4.4 for the evaluation of cyclotron emissivity).The direction of propagation of seed photons is determined by using a uniform random number, assuming isotropic emission in the fluid rest frame. After the direction is determined, the frequency and the direction of each seed photon are converted to those for the observer frame via the Lorentz transformation, by using the fluid velocity at the point where the seed photon is generated.
• 3.  
  We solve the photon transport problem taking into account free–free absorption, photon trapping (Begelman 1978; Ohsuga et al. 2002, 2003), and thermal/bulk Comptonization.

Each photon packet is given an initial weight w = 1. Along the photon trajectory between scatterings, the weight of the photon packet is reduced by the escape to the observer, by being swallowed by the black hole, and by free–free absorption. We note that the word "escape" is used in referring to "leak" in this paper. Due to the escape from the calculation box, the weight of photon packets is reduced for each scattering by a fraction 1 − P_esc, where P_esc  ≡  exp (− τ_esc) is the escape probability (i.e., the probability that photons will escape to observer or will be swallowed by the black hole without scattering or absorption). The optical depth τ_esc is obtained by integrating the opacity of the free–free absorption and the electron scattering from the current position of the photon packet to the outer boundary or to the inner boundary at r = 2r_s, along its current direction. When the ray crosses the inner boundary, the fraction P_escw of the photons is regarded as swallowed by the black hole. This mimics the effects of photon trapping because we take into account the beaming of photons due to the fluid motion. In the other cases, the fraction P_escw of the photons is summed up as observable photons. In addition to the reduction due to the escape of photons, the free–free absorption decreases the weight of photon packets by a fraction 1 − P_abs, where P_abs is the absorption probability determined by the optical thickness for free–free absorption between the current and the next scattering positions, that is, P_abs = 1 − exp (− τ_abs).

In order to obtain P_esc, we calculate τ_esc determined by

$$\begin{eqnarray} {\tau }_{\rm esc} &=& \sum ^{N}_{k=1}{\Delta }{\tau }(k), \end{eqnarray} \tag{ 2 }$$

where k is the sequential number of the grid cells along the ray counted from the current position of the photon packet, N is the total number of grid cells which the ray crosses, and Δτ(k) is the optical thickness of the kth cell computed by integrating the effective extinction coefficient α_{ν, eff} = α^ff_{ν, eff} + α^KN_{ν, eff} along the segment of the ray inside the kth grid cell, where α^KN_{ν, eff} is the effective scattering coefficient corresponding to the Klein–Nishina cross-section which is modified by the fluid motions and thermal motions of electrons (see Appendix B for detail).

Here, α^ff_{ν, eff} is the effective absorption coefficient modified by fluid motions only, since the modification of absorption cross-section due to thermal motion of ions (or electrons for the electron–electron bremsstrahlung) is already taken into account when we obtain the absorption coefficient from emissivity by using Kirchhoff's law. The extinction coefficients inside the cell are evaluated by linearly interpolating the coefficients at eight grid points at the corner of the cell.

By using the P_esc obtained by calculating τ_esc, we store the fraction of escaping photons in the packet as wP_esc for each energy bin in order to calculate the observable spectra (or spectra of photons which are swallowed by the black hole), and reduce the weight of the photon packet remaining in the computational domain as w(1 − P_esc).

In order to calculate the position for the next scattering event and the reduction of the weight of the photon packet due to absorption, we store the optical depth for scatterings from the current position ${\bm x}(n=1)$ toward the kth cell τ^KN(k) = ∑^k_{n = 1}Δτ^KN(n), that for absorption τ^abs(k) = ∑^k_{n = 1}Δτ^abs(n), and the position ${\bm x}(k)$ where the ray enters the k th cell. Here, Δτ^KN(n) and Δτ^abs(n) are the optical thickness for scattering and the absorption of the nth cell computed by integrating α^KN_{ν, eff} and α^ff_{ν, eff}.

The next scattering point is determined by computing the optical depth where the next scattering will occur, which is determined by ${\tau }_{\rm scat} = -{\ln } [1-{\xi } (1- e^{-{\tau }_{\rm esc}^{\rm KN}})]$. Here ξ is a uniform random number in the range (0, 1) and τ^KN_esc is the optical depth for the Compton scattering between the current position of the photon and the boundary (i.e., τ^KN_esc = τ^KN(N)). By finding the kth cell along the ray which satisfies τ^KN(k)  ⩽  τ_scat < τ^KN(k + 1), the next scattering point ${\bm x}_{\rm scat}$ is obtained by

$$\begin{equation} {\bm x}_{\rm scat} = \frac{{\tau }_{\rm scat} - {\tau }^{\rm KN}(k)}{{\tau }^{\rm KN}(k+1) - {\tau }^{\rm KN}(k)} [{\bm x}(k+1) - {\bm x}(k)] + {\bm x}(k). \end{equation} \tag{ 3 }$$

The optical depth for absorption between the current and the next scattering positions τ_abs is obtained from τ_abs(k), which is stored when we calculated τ_esc described above. We evaluate the absorption probability P_abs = 1 − exp (− τ_abs) and reduce the weight of the photon packet by a fraction (1 − P_abs). The weight of the photon packet w(1 − P_esc)(1 − P_abs) is available for the next scattering.

After the position of the next scattering is determined and the weight of photon packet remaining in the calculation box is evaluated, we calculate the energy shift of the photon packet due to thermal and bulk Compton scatterings. We obtain the electron temperature and the fluid velocity at that position by linear interpolation from those at the eight grid points at the corners of the cell including the scattering point. The details of the algorithm of the thermal Compton scattering are described in Pozdniakov et al. (1977) and Pozdnyakov et al. (1983). We incorporate the bulk Comptonization by transforming the photon four-momentum from the observer frame to the comoving frame of the fluid before the thermal Compton scattering. We convert the four-momentum into the observer frame after the Compton scattering.

For a frequency range of 10¹⁶ Hz < ν < 10²² Hz, we divide the photons into 50 frequency bins with Δlog ν = constant. We generate 4 × 10⁵ seed photons for each frequency bin and apply the above method to calculate their trajectories. These steps are repeated for photons belonging to the same frequency bin. We obtain the spectral energy distribution (SED) by summing up the energy of the escaping photons multiplied by the photon weight.

We have shown that the Comptonized SEDs are harder than those of a slim disk model (e.g., Watarai et al. 2005) by post-processing the results of the radiation hydrodynamic simulations of supercritical accretion onto a black hole. The X-ray spectrum of mildly supercritical accretion flows (${\dot{M}}$  ≈ 200 L_E/c²) has a power-law component in 1–10 keV and a rollover appears around 5 keV in highly supercritical accretion flows (${\dot{M}}$ ≳ 500 L_E/c²).

Figure 6 schematically shows the structure of Comptonizing supercritical accretion flows and the photon trajectories. Hard X-ray photons are produced near the black hole. Photons generated in the disk are scattered many times and are diffusively transported to the black hole because the beaming due to accreting gas motion helps photons to move toward the black hole. The sub-relativistic accretion near the black hole makes the photon frequency increase by the bulk Compton scattering. The photons are significantly upscattered by the thermal Compton scatterings at the shock-heated region. Some photons which are upscattered near the black hole are injected into the funnel jet. Some of them escape along the rotation axis. The others are scattered by electrons in the funnel jet and injected into the cool outflow surrounding the funnel jet. They are downscattered by the thermal electrons in the cool outflow. Some fraction of the downscattered photons escape to the observer. The soft photons (≲1 keV) are produced and self-Comptonized in the cool outflow. We note that most photons upscattered near the black hole are swallowed by it.

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The spectral softening due to thermal electrons in the cool outflow becomes more significant when ${\dot{M}}$ is higher, because the narrower and denser funnel jet decreases the number of photons which escape without entering cool outflow surrounding the jet. When ${\dot{M}}$ ≈ 200 L_E/c², the SED is significantly hardened by thermal Comptonization in a shock-heated region as shown in Figure 5. In contrast, when the mass accretion rate is higher (${\dot{M}}$  ≈ 1000 L_E/c²), the SED calculated by taking into account the thermal Comptonization only becomes softer than that calculated without Comptonization, because the downscatterings in the cool outflow overwhelm the upscatterings in the shock-heated region. When we take into account the bulk Comptonization by the sub-relativistic outflow, the outflow motion in the funnel assists photons to escape without significant downscattering and the SED is eventually hardened.

When the mass accretion rate is higher, our calculations in this paper show that SEDs become softer, although our previous paper (Kawashima et al. 2009) expected that SEDs become harder because of Compton upscattering by thermal electrons in the jet. Kawashima et al. (2009) evaluated the photon indices by calculating the Compton y-parameters in the region τ_eff ⩽ 1. The effective optical depth is calculated by integrating the free–free absorption and the scattering coefficients from the outer boundary at r = 500r_s to the inner boundary at r = 3r_s along the rays with polar angle θ = constant, for various θ. (Here, we note that the computational domain for radiation hydrodynamic simulation in Kawashima et al. 2009 was smaller than that in this work.) Since Kawashima et al. (2009) did not take into account the downscattering by cool outflows, they overestimated the spectral hardening when the accretion rate is high. When the mass accretion rate is higher, a greater number of escaping photons tends to enter into the cool outflow surrounding the jet and they are subsequently downscattered, because the funnel becomes narrower and jet becomes thicker for Thomson scatterings. This is the reason why the SEDs become softer when the mass accretion rate is higher, in this work. Nevertheless, the conclusion in Kawashima et al. (2009) that the spectral state changes from the slim disk one to the Comptonizing outflows is valid in highly supercritical accretion flows because the SED of the photons escaping from such highly supercritical flows is determined by the Compton downscatterings in the outflow.

Recent X-ray observatories (Suzaku, XMM-Newton, and Chandra) provide us with precise spectra of ULXs below 10 keV. In Figure 7, we compare the SEDs obtained in this work and the XMM-Newton pn data of three ULXs: NGC 1313 X-2, IC 342 X-1, and NGC 5204 X-1, which are kindly provided by J. C. Gladstone. The SEDs of our results are normalized by the X-ray luminosity shown in Figure 3, while the SEDs of ULXs are normalized by the X-ray luminosity shown in Gladstone et al. (2009). The SEDs obtained by our numerical simulations are similar to those of ULXs. The observed X-ray spectra of ULXs with photon index Γ ≲ 2 can be fitted well by the models with i ≲ 30°. On the other hand, for ULXs with photon index Γ ≳ 2, spectra with 20° ≲ i ≲ 50° are fitted well.

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For IC 342 X-1 and NGC 5204 X-1, the soft X-ray component below 1 keV of our results is weaker than that of the ULXs. This is because the contribution from the cool outflow in the region R ≳ 100r_s is underestimated because the temperature of the outflow might be too cool to significantly contribute to the SED at ≲1 keV or our computation box might be too small. We might be able to resolve this discrepancy by carrying out MHD simulations taking into account the dissipation of magnetic energy in the disk corona and in the outflow, which can heat the outflow. In addition, the magnetic fields affect the geometrical structure of jets. Takeuchi et al. (2010) performed axisymmetric two-dimensional radiation MHD simulations and found that the jet is collimated by magnetic stress, while it is accelerated by radiative force. Therefore, the structure of the jet and the outflow is affected by magnetohydrodynamical processes. It will be our future work to calculate photon spectra by using the results of radiation MHD simulations.

Let us compare the luminosity obtained from our simulations and ULXs. It should be noted that our simulation assumed a black hole with M = 10 M_☉. The luminosity increases or decreases according to the black hole mass. The X-ray luminosity for the models with ${\dot{M}}$  ≈ 1000L_E/c² and i = 10°–20° is ∼10⁴⁰ erg s⁻¹ (Figure 7), while that of NGC 1313 X-2 evaluated in Gladstone et al. (2009) is ∼5 × 10³⁹ erg s⁻¹. Since the X-ray spectrum of NGC 1313 X-2 is well fitted by our simulation results with these parameters for the accretion rate and the viewing angle, the luminosity difference suggests that NGC 1313 X-2 may harbor a black hole whose mass is 5 M_☉, since the luminosity of accretion flows is proportional to the mass of black holes when ${\dot{M}}/(L_{\rm E}/c^{2})$ is constant. On the other hand, the X-ray luminosity for the models with ${\dot{M}}$ ≈ 200L_E/c² and i = 40°–50° is ∼3 × 10³⁹ erg s⁻¹, while that of NGC 5204 X-1, whose spectrum in the 1–10 keV band is well fitted by this accretion rate and viewing angle in our simulations, is ∼5 × 10³⁹ erg s⁻¹. This indicates that NGC 5204 X-1 possesses a slightly massive black hole with mass M ∼ 20 M_☉. We leave it as a future work to carry out radiation hydrodynamical (or radiation magnetohydrodynamical) simulations and spectral calculations for supercritical accretion onto a black hole whose mass is smaller or larger than 10 M_☉.

Recently, T. Yoshida et al. (in preparation) confirmed that IC 342 X-1 shows power-law spectra with Γ ∼ 1.6–1.8 extending up to ∼20 keV during the lower luminosity phase in Suzaku observations, while it shows power-law spectra with photon index Γ ∼ 1.8–2.0 accompanying a rollover at ∼5 keV during the higher luminosity phase in some XMM-Newton observations. In our results, for viewing angle i = 10°–20°, the power-law component extends up to 10 keV in a lower ${\dot{M}}$ model, while the rollover at ∼5 keV appears in a higher ${\dot{M}}$ model. Furthermore, each photon index of power-law components obtained from our numerical simulations is consistent with that of X-ray observations. This suggests that supercritical accretion occurs in IC 342 X-1.

We note that the SEDs for the model with low mass accretion rate (${\dot{M}}$  ≈ 200L_E/c²) are similar to those of galactic black hole candidates in the very high state, except that obtained SEDs possess a harder power-law component. Our calculations indicate that even when the ULXs show SEDs similar to that of the very high state, whose luminosity is L  ≲  L_E, we cannot conclude that the ULXs are subcritically accreting IMBHs because supercritically accreting stellar mass black holes can also produce SEDs similar to the very high state.

In this study, we ignored the cyclotron and synchrotron emission. As we mentioned in Section 2, this is partly because our radiation hydrodynamic simulation does not treat the effect of magnetic field, and partly because the disk (or the outflow covering the outer disk) is dense and hot enough to produce the radiation field reaching blackbody via only the bremsstrahlung emission processes in the soft X-ray band.

Here, we show that the cyclotron emission might not contribute to the X-ray spectra. If we apply E_mag ∼ 10E_gas from the radiation MHD simulations (Ohsuga & Mineshige 2011) to the result of our radiation hydrodynamic simulation, we obtain B ∼ 10⁴ G in the funnel jet, where E_mag and E_gas are the magnetic and the gas energy density, respectively. The physical condition in the funnel jet (ρ ∼ 10⁻⁹g cm⁻³, T_gas ∼ 10⁸ K, and B ∼10⁴ G) is close to one examined by Takaraha & Tsuruta (1982), which shows that the emission of self-absorbed cyclotron higher harmonics becomes optically thin when ν  ≳ ν_* = 3 × 10¹² Hz in that condition, where ν_* is the critical frequency. The self-absorbed cyclotron–synchrotron intensity has a peak in I_ν = 2ν²_*k_BT_gas/c² at ν = ν_* (∼3 × 10¹² Hz, i.e., far-IR band). This peak intensity due to self-absorbed cyclotron emission from the jet is lower than that of the blackbody intensity in the UV or soft X-ray band arising from the disk (T_gas ∼ 10⁷ K). Since the intensity due to cyclotron–synchrotron emission in the UV or soft X-ray band is much less than that of the peak intensity, it is also less than that of the blackbody emission from the disk. This suggests that the seed photons contributing to the Comptonized X-ray spectra are dominated by the bremsstrahlung emission from the disk rather than the cyclotron emission from the jet.

We note that the cyclotron emission from the shock-heated region is also too weak to affect the soft X-ray spectra. In the shock-heated region, since the estimated magnetic field is B ∼ 10⁶ G and the critical frequency is ν_*∼ 10¹⁴ Hz, i.e., near-IR band, the corresponding peak intensity is still lower than that of ∼10⁷ K blackbody in the UV or soft X-ray band. It does not affect the X-ray spectra even if they are strongly self-Comptonized.

The cyclotron emissions and their Compton upscatterings are expected to be important in frequencies from the radio to the visible band. For the calculation of spectra from the radio to hard X-ray band including the cyclotron–synchrotron emission, it is important to carry out the radiation MHD simulations in order to obtain the structure of magnetized supercritical accretion flow. This remains as a future work.

Our spectral calculations do not incorporate the general relativistic effects. By the gravitational redshift, the energy of photons Comptonized up to the hard X-ray band by heated electrons at r  ≲  5r_s will be decreased by 10%–30% in Schwarzschild spacetime. Subsequently, the hard X-ray tail would become softer than that of the calculation for flat spacetime. Niedźwiecki & Zdziarski (2006) calculated the photon spectra Comptonized by spherically free-falling accretion flow, whose seed photons are emitted from the accretion disk outside the spherical accretion flows (but sometimes the disk extended down to the black hole) in flat, Schwarzschild, and Kerr spacetime. They compared the results for flat spacetime and those for Schwarzschild coordinates, and found that the effect of gravitational redshift softens the photon spectra in the hard X-ray band. Their results indicate that the difference between the spectra for the flat and the Schwarzschild spacetime becomes small in significant supercritical accretion model ${\dot{M}}$ = 12 L_E/c² because most of the photons scattered up to the hard X-ray band by the bulk Compton effect are swallowed by the black hole via the photon trapping effect. Although this seems to indicate that the general relativistic treatment does not affect the observable SEDs, it might not be negligible because our results show that the funnel jet assists photons, which are upscattered in the shock-heated region near the black hole, to escape to observers. The spectral calculation of supercritical accretion flows taking into account the general relativistic effect remains as a future work.

We emphasize that the funnel jet affects the dependence of the observable spectra on viewing angle. Niedźwiecki & Zdziarski (2006) showed that the photon spectra become softer when the viewing angle is lower, because the soft photons are emitted from the geometrically thin accretion disk and the isotropic high-energy photons arise from the spherical accretion flows via Compton scatterings. On the other hand, our result shows that the radiation spectra become harder when the viewing angle is smaller (Figure 4), because the hard X-ray photons escape through the funnel (see Figure 6). We conclude that it is important to take into account the disk geometry for spectral calculation of accretion flows.