EMERGENT SPECTRA FROM DISKS SURROUNDING KERR BLACK HOLES: EFFECT OF PHOTON TRAPPING AND DISK SELF-SHADOWING

The basic equations governing the disk structure used in our work are similar to those given in Shimura & Manmoto (2003). In the Boyer–Lindquist coordinates, expanding around the equatorial plane up to (z/r)⁰ terms, the metric takes the form (Novikov & Thorne 1973, the geometric unit G = C = M = 1 is taken)

$$\begin{equation} ds^{2}=-r^{2}\frac{\Delta }{A}dt^2+\frac{A}{r^2}(d\phi -\omega dt)^2+\frac{r^2}{\Delta }dr^2+dz^2, \end{equation} \tag{ 1 }$$

with

$$\begin{equation*} \Delta =r^2-2r+a^2,\end{equation*}$$

$$\begin{equation*} A=r^4+r^2 a^2+2 r a^2,\end{equation*}$$

$$\begin{equation*} \omega =\frac{2 a r}{A},\end{equation*}$$

where a is the dimensionless spin of the black hole.

The mass conservation takes form

$$\begin{equation} \dot{M}=2 \pi \Delta ^{1/2}\Sigma \gamma _{r} V, \end{equation} \tag{ 2 }$$

where $\dot{M}$ is the mass accretion rate, and W and Σ are integrated pressure and density, defined as $W=\int p \mathrm{\mbox{\mbox{\it d}}}z$, $\Sigma =\int \rho \mathrm{\mbox{\mbox{\it d}}}z$. Here, p and ρ denote pressure and density, respectively, the subscript "0" denotes the corresponding value at the equatorial plane, and H is the scale height of the accretion disk. γ_r is the radial Lorentz factor of the fluid.

The angular momentum conservation reads

$$\begin{equation} \frac{\dot{M}}{2\pi }(L-L_{\rm in})=\alpha \frac{A^{3/2}\Delta ^{1/2} \gamma _{\phi }^3}{r^5} W, \end{equation} \tag{ 3 }$$

where L is the specific angular momentum, L_in is the specific angular momentum at the inner boundary of the accretion flow, and γ_ϕ is the Lorentz factor in the ϕ direction as defined in Manmoto (2000). As the viscosity prescription used here is conventional (Narayan 1992), at the inner part of the accretion disk, our result may be inaccurate in principle. However, the causal viscosity is still somewhat putative and differs from the conventional one only close to or inside the ISCO. Thus, the disk spectrum in our Kerr case is robust in the sense that the sonic point is very close to the event horizon of the black hole, while the disk spectrum in the Schwarzschild case may be shifted a bit due to the different treatment of the prescription of the viscosity.

The momentum equation and energy equation take the forms as follows.

• 1.  
  Momentum equation:

  $$\begin{equation} \gamma _{\it r}^2 V \frac{\mathrm{\mbox{\mbox{\it d}}} V}{\mathrm{\mbox{\mbox{\it d}}} r}=-\frac{1}{\Sigma }\frac{\rm \mbox{\mbox{\it d}} W}{\mathrm{\mbox{\mbox{\it d}}} r}-\frac{\gamma _{\phi }^2 A}{r^4\Delta } \frac{\big(\Omega -\Omega _{\rm k}^+\big)(\Omega -\Omega _{\rm k}^-)}{\Omega _{\rm k}^+\Omega _{\rm k}^-}, \end{equation} \tag{ 4 }$$

  where

  $$\begin{equation*} \Omega _{\rm k}^{\pm }=\pm \frac{1}{r^{\frac{3}{2}}\pm a}\;\end{equation*}$$

  is the Keplerian angular velocity of the direct ("+ sign") and retrograde ("−" sign) circular orbit, respectively. And here

  $$\begin{equation*} \gamma _{\phi }=\Big (1+\frac{r^2 L^2}{\gamma _{\it r}^2 A}\Big)^{1/2}\;\end{equation*}$$

  is Lorentz factor of v_ϕ

  $$\begin{equation*} \Omega =\omega +\tilde{\Omega }=\omega +\frac{r^3 \Delta ^{1/2} L}{A^{3/2} \gamma _{\it r}\gamma _{\phi }}.\end{equation*}$$
• 2.  
  Energy equation:

  $$\begin{equation} Q^+_{\rm vis}=Q^-_{\rm adv}+F_{\rm rad}^-, \end{equation} \tag{ 5 }$$

  here, Q⁺_vis is the viscous heating rate per unit surface, F⁻_rad is the radiative cooling term, and Q⁻_adv is the effective cooling induced by heat advection. They can be specified as follows:

  $$\begin{equation} Q^+_{\rm vis}=-\frac{1}{4 \pi r}\frac{\gamma _{\phi } A^{1/2}}{\Delta ^{1/2} r} \dot{M} (L-L_{\rm in})\frac{\mathrm{\mbox{\mbox{\it d}}} \Omega }{\mathrm{\mbox{\mbox{\it d}}} r}, \end{equation} \tag{ 6 }$$

  $$\begin{eqnarray} Q^-_{\rm adv}&=& -\frac{\dot{M}}{4 \pi r}\frac{W}{\Sigma }\frac{1}{\Gamma _3-1}\bigg (\frac{\mathrm{\mbox{\mbox{\it d}}} \ln W}{\mathrm{\mbox{\mbox{\it d}}} r}\nonumber \\ &&-\Gamma _1 \frac{\mathrm{\mbox{\mbox{\it d}}} \ln \Sigma }{\mathrm{\mbox{\mbox{\it d}}} r} -(\Gamma _1+1)\frac{\mathrm{\mbox{\mbox{\it d}}} \ln H}{\mathrm{\mbox{\mbox{\it d}}} r} \bigg), \end{eqnarray} \tag{ 7 }$$

  where Γ₁ and Γ₃ are defined in Kato et al. (1998), and

  $$\begin{equation} F^-_{\rm rad}=\frac{8 a c T^4}{3 \kappa \rho _0 H}. \end{equation} \tag{ 8 }$$

  The Rosseland mean of total opacity can be expressed as

  $$\begin{equation*} \kappa =0.4+0.64\times 10^{23}\rho T^{-7/2}.\end{equation*}$$

The total pressure can be expressed as the sum of the gas and radiation pressure. For simplicity, we drop the numerical factors in Shimura & Manmoto (2003), and the equation of state is given by

$$\begin{equation} W=\frac{1}{3}a T^4 H+\frac{2 k_{\rm B}}{m_{\rm p}}\Sigma T, \end{equation} \tag{ 9 }$$

where T is temperature of the disk. Another equation is the vertical thickness equation, which reads

$$\begin{equation} H=\sqrt{12}\frac{c_{\rm s}}{\Omega _{\rm k}}g_{\rm pa}^{-\frac{1}{2}}. \end{equation} \tag{ 10 }$$

This is a modification to the equations of Shimura & Manmoto (2003), and its derivation will be presented in Section 2.2.

The above equations are transformed into a set of first-order differential equations with variables v_r and W, and then integrated from the outer boundary at about 10⁵r_g.

The transonic solutions (Liang & Thompson 1980) are obtained using a shooting method, in which we adjust L_in recursively until the smooth transonic solution is found. During the solving process, despite that the factor g_pa takes form

$$\begin{equation} g_{\rm pa}^{\prime }=\gamma _{\phi }^2\bigg (\frac{(r^2+a^2)^2+2 \Delta a^2}{(r^2+a^2)^2-\Delta a^2}\bigg), \end{equation} \tag{ 11 }$$

a simplified version of factor g_pa is used when finding the global transonic solution:

$$\begin{equation} g_{\rm pa}=\frac{(r^2+a^2)^2+2 \Delta a^2}{(r^2+a^2)^2-\Delta a^2}. \end{equation} \tag{ 12 }$$

However, the location of the photosphere (Peitz & Appl 1997) is still determined according to Equation (13). When this simplification is taken, at the inner regions of the accretion disks, the resulting disk photosphere and disk thickness no longer agree with each other. But the difference is not significant. We thus neglect this inconsistency and use the location of the disk photosphere in the ray-tracing calculation.

The vertical thickness is a major uncertainty in the accretion disk theory and is claimed to be dependent on various physical assumptions such as the vertical dissipation profiles and the interplay between the relaxation time scale and the free-fall time scale. However, Sadowski et al. (2009) showed that the vertical thickness of an accretion disk in the non-advective regime can be approximated using this simple formula

$$\begin{equation} \kappa \sigma T_{\rm eff}^4\frac{1}{c}=\frac{2}{3} \Omega _{\rm k}^2 z g_{\rm pa}, \end{equation} \tag{ 13 }$$

where κ is the total opacity, σ is Stefan–Boltzmann constant, and z is the photosphere location. T_eff is the effective temperature defined as T_eff = (F⁻_rad/σ)^1/4 and Ω_k = r^−3/2. Equation (13) simply means that at the disk photosphere, the vertical radiation pressure balances the gravity of the central black hole. In this work, despite the fact that Equation (13) has only been shown to be valid in the non-advective regime, we also use it in the advective regime, since the physical argument is still valid. With this assumption, we can get the expression for the location of the photosphere:

$$\begin{equation} z=\kappa \frac{3 \sigma T_{\rm eff}^4 }{2 \Omega _{\rm k}^2 g_{\rm pa}}\frac{1}{c} =\frac{3}{2}\kappa \frac{F_{\rm rad}}{c}\Omega _{\rm k}^{-2} g_{\rm pa}^{-1}. \end{equation} \tag{ 14 }$$

Another constraint comes from the vertical radiation flux. According to the diffusion approximation, assuming the dominance of the radiation pressure, the radiation flux can be approximated as

$$\begin{equation} F_{\rm rad}=\frac{8 a c T^4}{3 \kappa \rho _0 H}=\frac{8 c W}{\kappa \Sigma H}. \end{equation} \tag{ 15 }$$

We thus make the physical requirement that the thickness z obtained in Equation (14) should be consistent with the thickness H in Equation (15). Combining these two equations, we obtain the expression for modified disk thickness:

$$\begin{equation} H=\sqrt{12}\frac{c_{\rm s}}{\Omega _{\rm k}}g_{\rm pa}^{-\frac{1}{2}}. \end{equation} \tag{ 16 }$$

It should be noted that Equation (16) is accurate when the disk is radiation pressure dominated. At the outer regions of the accretion disks, where the gas pressure dominates, Equation (16) deviates from the previous one by a factor of $\sqrt{12}$ (the previous expression takes form H = c_s/Ω_kg^−1/2_pa). However, as Sadowski et al. (2009) showed, the thickness in the outer region of the accretion disks is not well constrained, and it is dependent on the dissipation profile in the vertical direction. As the contribution of radiation from the gas-dominated region is relatively unimportant and the final emergent SED is unaffected by our modifications significantly, we use Equation (16) throughout the paper. One concern about the estimation of thickness is that the effect of magnetic field may influence the disk thickness. Hirose et al. (2006) pointed out that the location of photosphere of the accretion disk is modified by the magnetic field significantly. However, in our case, the shadowing effect happens at the inner region, where the radiation pressure dominates. Thus, the thickness and the self-shadowing might not be significantly affected by the magnetic effect. The magnetic field might also change the inner boundary condition significantly (e.g., Penna et al. 2010; Noble et al. 2010, and references therein). To investigate the effects of the magnetic field, we should do the general relativistic MHD simulations (Gammie et al. 2003; De Villiers et al. 2003), which is beyond the scope of this work.

After obtaining the global structure of the relativistic disk, we use the ray-tracing technique to calculate the emergent spectra with inclusion of all the relativistic effects. The ray-tracing method is based on the analytical integration of the null geodesics (Chandrasekhar 1983; Rauch & Blandford 1994; Cadez et al. 1998), and the subroutines are the same as those used in Yuan et al. (2009).

One major improvement compared with earlier work (e.g., Li et al. 2005) is that we include the disk thickness in the ray-tracing calculation. To achieve this, we start from the infinity and then search along the each photon trajectory for the intersection point.

Once the intersection points are found, we can calculate the local temperature of the emitting points in our disk model and calculate the redshift factor g. The emergent spectra are then obtained by integrating over the photometric plane according to

$$\begin{equation} F_{\nu _o}=\int g^{3} I_{\nu _{e}} \frac{\mathrm{d} \alpha \mathrm{d} \beta }{D^2}, \end{equation} \tag{ 17 }$$

where α and β are two impact parameters, and the redshift factor is

$$\begin{equation} g=\frac{\nu _{\rm o}}{\nu _{\rm e}}, \end{equation} \tag{ 18 }$$

here, D is the distance of the object expressed in geometrical units and I_ν takes the diluted blackbody form (e.g., Shimura & Takahara 1995).

The redshift factors are calculated as follow:

$$\begin{equation} g=\frac{\nu _{\rm obs}}{\nu _{\rm em}}=\frac{\left.p_{\mu } u^{\mu }\right|_{r=\infty,\theta =\theta _{\rm \rm obs}}}{\left. p_{\mu }u^{\mu }_{\rm Fluid}\right|_{r=r_{\rm em}\, \theta =\theta _{\rm em}} }, \end{equation} \tag{ 19 }$$

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

$$\begin{equation} p_{\mu }=(-1,\pm \frac{\sqrt{R}}{\Delta },\pm \Theta ^{1/2},\lambda). \end{equation} \tag{ 20 }$$

The four-velocity of the fluid is given by Yuan et al. (2009)

$$\begin{equation} u^{\mu }_{\rm Fluid}=(\gamma _{r} \gamma _{\phi } e^{-\nu },\gamma _{r}\beta _{\rm r} e^{-\mu _1},\ 0,\ \gamma _{r}\gamma _{\phi }(\omega e^{-\nu }+\beta _{\phi } e^{-\psi })), \end{equation} \tag{ 21 }$$

where e^ν = (ΣΔ/A)^1/2, e^ψ = (Asin²θ/Σ)^1/2. Thus, we have the expression for redshift factor:

$$\begin{equation} g=\frac{\Sigma ^{1/2}\Delta ^{1/2}}{A^{1/2}}\frac{1}{\gamma _r \gamma _{\phi }}\frac{1}{1\mp \frac{\beta _r}{\gamma _{\phi }}\frac{R^{1/2}}{A^{1/2}}-\lambda \omega -\lambda \beta _{\phi } \frac{r^2 \Delta ^{1/2}}{A}}, \end{equation} \tag{ 22 }$$

where "em" means the emitting points, and the "+" sign for the trajectories of the photons without r-turning point and "−" sign for those with r-turning point. In Equation (20),

$$\begin{equation} R=r^4+(a^2-\lambda ^2-Q) r^2+2(Q+(\lambda -a)^2)r -a^2 Q, \end{equation} \tag{ 23 }$$

here, the definitions of λ and Q are the same as that of Cunningham & Bardeen (1973).

In the ray-tracing calculations, the outer boundary condition of the integration is set at 10⁵r_g. In the cases when the emitting radius is so large that the disk solutions do not cover, we use the results from SSD instead, because the differences between the transonic disk model and SSD are indeed insignificant at the large radii.

One limitation in our calculation is that the returning irradiation (van Paradijs & McClintock 1995; Li et al. 2005) is neglected. In the inner regions of the accretion disk, due to the strong gravity, the radiation emitted at one part of the accretion disk may reach other parts, which may result in a modification to the disk structure and the emergent spectra. Certain disk geometry may enhance such kind of effect (van Paradijs & McClintock 1995). In the case of the thin accretion disk, this effect has moderate influence (about several percents) on the emergent spectra (Li et al. 2005). In our case, due to the thick geometry, since the disk is shadowed by itself, the self-irradiation effect may be stronger. As the inclusion of that effect requires significant amount of calculation, for simplicity, we neglect it in our current work.

Another limitation in our calculation is that the effect of limb darkening has not been considered. As Li et al. (2005) have pointed out, the limb darkening has effect on the emergent spectra, especially when the inclination angle is large, and the deviation induced by this effect is about 10% percent at low energy band when the disk is viewed at 85° (Li et al. 2005). In this work, this effect has not been included. According to the limb-darkening law I_θ ∼ 1/2 + 3/4cos θ, our results are still robust at relatively small inclination angles.

Figures 1–3 show the dynamical properties of our accretion disks. The upper panels in Figure 1 show the radial velocity as a function of radii, with the different accretion rates. Because we use fully relativistic equations, near to the event horizon of the black hole, as expected, the radial velocity is very close to the speed of light. At the small radii, for the same accretion rate, the higher the spin, the smaller the radial velocity. At the large radii, the radial velocity of the accretion flow is relatively small, and it is insensitive to the black hole spin. This is because that at the large radii, the disk structure resembles that of SSD. With the increase of the accretion rate, the radial velocity becomes larger.

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The lower panels of Figure 1 show the radial distribution of the surface density under the different accretion rates. Generally speaking, at the small radii, the surface density is small. These results can be simply understood in terms of the non-relativistic form of the matter conservation:

$$\begin{equation*} \dot{M} =2 \pi r \Sigma v_{\rm r},\end{equation*}$$

where Σ is the surface density of the accretion disks and v_r is the radial velocity (the relativistic one is similar but with an additional correction related to the radial Lorentz factor γ_r and the black hole spin a). Since the accretion disks around Schwarzschild black holes tend to have higher radial velocity, their surface density at given radius is smaller. Another noticeable feature is that the density hop occurs when $\dot{M}=0.25 \dot{M}_{\rm edd}$, a = 0, which is accompanied by the velocity decrease and re-increase in the upper panels. This peculiar behavior is not shown in Shimura & Manmoto (2003) since they did not explore the disk solution at such a low accretion rate, but can be found in the previous papers solving the non-relativistic slim disk structure (Chen & Wang 2004) and a recent relativistic one (Sądowski 2009, with explanation).

Upper panels of Figure 2 show the rotation velocity of the accretion flow. The rotation velocity is insensitive to the accretion rate, especially at the outer region. To show how the fluid motion deviates from the Keplerian motion, the ratio of the rotational velocity to the rotational velocity of the Keplerian motion v_ϕ/v^Kep_ϕ is shown in the lower panels of Figure 2. When the accretion rate is low, the rotation velocity ratio is very close to unity, which means the assumptions made in SSD are accurate. When the accretion rate is relatively high, the rotation of the fluid becomes sub-Keplerian. In some regions, the rotation is super-Keplerian. Although there exist some differences, the deviation is no more than 10% at several gravitational radii, which means the assumption of the Keplerian motion is still valid at the most of the regions of the accretion disks (see, e.g., Igumenshchev & Abramowicz 2000).

Figure 3 shows the effective temperature T_eff as a function of radii. Here, T_eff is defined as T_eff = (F⁻_rad/σ)^1/4, where F⁻_rad is the radiation flux per unit area and σ is the Stefan–Boltzmann constant. Similar to the above figures, the differences between the Schwarzschild and the Kerr black holes are all very small at large radii. With the increase of the accretion rate, the temperature increases. When the accretion rate is high enough, the temperature is still high even inside the ISCO, which suggests that, contrary to the assumptions made in the SSD model, the matter inside the ISCO is not cold.

Figure 4 shows the thickness of the disk versus the radii. In order to give a instinctive picture of the disk thickness, we use linear coordinates and show only the results in the inner part of the accretion disk. This thickness profile is thus correspond to the surface of the accretion disk. The imprints of ISCO appear in the disk thickness profile. This feature is particularly prominent at low accretion rate, where the disk surface agrees with the simulation carried out by Hawley & Krolik (2002) and Reynolds & Fabian (2008). When the accretion rate is high enough, the imprints of ISCO disappear, and the disk shows a continuous profile down to the event horizon of the black hole.

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However, in some cases, the estimation of thickness may be incorrect. Since the estimation of the thickness is based on the vertical hydrostatic equilibrium, it depends on the assumption:

$$\begin{equation*} a_{\rm z}^{\rm Fluid}<g_{\rm z}, \end{equation*} \tag{ 25 }$$

which means the acceleration in the vertical direction could never exceed the acceleration provided by the vertical gravitational force (or that the disk has time to relax to the hydrostatic equilibrium). We checked our solutions using this formula and found that in some cases such as $\dot{M}=0.25$, a = 0, Equation (25) is not satisfied. Thus, the disk thickness may be underestimated at the regions near ISCO, in some cases.

In Figure 5, we present our results on the global energy balance of the accretion disks. The energy conservation in the accretion disks takes the simple form as follows:

$$\begin{equation} Q_{\rm {vis}}^+=F_{\rm {rad}}^-+Q_{\rm {adv}}^-, \end{equation} \tag{ 26 }$$

where Q⁺_vis is the viscous energy generation rate, F⁻_rad is the cooling term due to radiation from the accretion disk, and Q⁻_adv is the effective cooling induced by the heat advection. If Q⁺_vis ∼ F⁻_rad, the disk is very similar to the standard accretion disk since the energy generated by the viscous processes is radiated locally. If Q⁺_vis ∼ Q⁻_adv, most of the heat generated is trapped inside the accretion disk.

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Figure 5 shows the ratio of the emitted radiation from the disk surface to the total heat generated by the viscous processes. At about 20 r_g, the heat advection works, as a result, the ratio is less than 1. However, in the inner region, in many cases, F⁻_rad dominates over Q⁺_vis, which means that substantial amount of energy is still being radiated away. The source of the energy mainly comes from the energy trapped in the region outside the ISCO. Thus, our results suggest that the radiation trapped outside the ISCO can be allowed to escape at the regions close to the black hole. This kind of mechanism is new, since different from the advective cooling, in some cases, the effect of advection will heat the disk up, resulting in "advective heating."

Besides from getting self-consistent disk solution, we find that when the accretion rate is low enough such as $\dot{M} \sim 1/4$ for a = 0 and $\dot{M} \sim 1/16$ for a = 0.98, the SSD model is valid.

It is of theoretical interest to investigate what influences the effect of the self-shadowing. Without the light-bending effect, the self-shadowing effect is dependent on the maximum of H/r, which is mainly dependent on the black hole accretion rate (McClintock et al. 2006). However, it remains unclear how the shadowing effect influenced by the strong gravity.

To gain physical insight into this question, we choose two systems with exactly the same luminosity but the different spin (one with $\dot{M}=1$ and a = 0.974, another with $\dot{M}=4$ and a = 0) and calculate the spectra. For both systems, we calculate the spectra both with and without the effect of light bending. In the latter case, for self-consistency, the redshift is evaluated as

$$\begin{equation} g=\frac{\Delta ^{1/2} r}{\gamma _{r}\gamma _{\phi }A^{1/2}}\frac{1}{1\pm \frac{\beta _{\rm r} \sqrt{R}}{\gamma _{\phi }A^{1/2}}-\Omega \lambda }. \end{equation} \tag{ 29 }$$

After that, to quantify the effect of disk self-shadowing, we compare the SED calculated with disk thickness and the SED calculated without disk thickness by defining the spectral distortion as function of frequency as

$$\begin{equation} {\rm Distortion(\nu)}=\frac{\rm SED_{\rm not\ shadowed}-\rm SED_{\rm shadowed}}{\rm SED_{\rm not\ shadowed}+SED_{\rm shadowed}}, \end{equation} \tag{ 30 }$$

where the $\rm SED_{\rm not\ shadowed}$ denotes the SED calculated by assuming slim disk with no thickness and $\rm SED_{\rm shadowed}$ denotes the SED calculated including the disk thickness.

The spectral distortion for different spin and different assumptions about light trajectories is plotted in Figure 10. For both the Schwarzschild and the Kerr black holes, including the light-bending effect brings about significant distortion to the emergent spectra, and this distortion generally appears as attenuation of the flux at high energy part. This kind of distortion is more significant for Kerr black hole, because the light trajectories are more severely influenced by the strong gravity.

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It was known that the spectral evolution of the X-ray binary deviates from the trend predicted in the SSD model at both the low and high luminosities (Done et al. 2007; Kubota & Makishima 2004). Although the deviation at the low luminosity states is often interpreted as the evaporation of the optically thick disk into the optically thin advection-dominated accretion flow (Meyer & Meyer-Hofmeister 1994), the deviation at the high luminosity states is still mysterious.

The standard accretion disk model predicts the monotonic evolution of the disk peak energy with the luminosity, i.e., the higher peak energy corresponds to the higher luminosity. This kind of spectral evolution is consistent with the spectra evolution of X-ray binary at the medium luminosity states. However, beyond some critical luminosity L_crit, the peak of the disk SED shifts to the low energy band (Done et al. 2007) with the increase of the disk luminosity. This kind of spectral evolution is difficult to understand in the SSD model.

Various attempts have been made to tackle this problem, which include a change of the color correction with the luminosity (Davis et al. 2006) as well as the effects of the optically thick advection. One common difficulty with these explanations is that they all predict a single critical luminosity L_crit, which is in contradiction with the diversity of the critical luminosity observed in the different sources (Done et al. 2007).

One natural explanation from our calculations is that the abnormality of the spectral evolution is due to the self-shadowing of the accretion disks. The trend of the observed spectral evolution is consistent with our results as shown in Figure 6, and the diversity of the critical luminosity L_disk/L_edd in the different sources might be due to the different inclination angle and black hole spin. According to our results, the critical luminosity becomes smaller for the larger inclination angle or the black hole spin.

Our results also provide a diagnostic of the location of the shadowing medium. If the shadowing happens at the inner region of the accretion disks, the critical luminosity is dependent on black hole spin as well as the inclination angle. If the disk is shadowed by the matter distant from the black hole, (e.g., equatorial wind; Done & Davis 2008), the critical luminosity is dependent only on the black hole spin.

The spin of GRS 1915+105 is debated in the literature. By fitting the disk blackbody component, McClintock et al. (2006) found that the spin is near extreme, while Middleton et al. (2006) found the spin is moderate, with a ∼ 0.8. One major difference between these two works is that the spin obtained by McClintock et al. (2006) is based on the spectra fitting for the relatively low luminosity (L < 0.3L_edd) states, while Middleton et al. (2006) set no strict restriction on the luminosity states. Interestingly, McClintock et al. (2006) found that the measured spin becomes smaller for the higher luminosity states.

To show how the spin can be recovered by the fitting procedure, we simulate the artificial data in our model which is added with comptt (the comptonization model in XSPEC), iron absorption edge, iron absorption line, as well as galactic absorption, and then we fit the data with the same procedure as used in McClintock et al. (2006). All the parameters are the same as those in McClintock et al. (2006). The only difference in the artificial spectral fitting is that we replace the KERRBB2 model with KERRBB model and set the spectral hardening factor to be 1.7.

Our result is shown in Figure 11. Without introducing the additional parameters, we are able to roughly reproduce the trend found in McClintock et al. (2006). At both the low (around 1/4 of Eddington luminosity) and high (close to 1 Eddington luminosity) luminosity states, our simulated result agrees well with the result of McClintock et al. (2006). When the disk luminosity is low enough, according to the spectral model we have developed, the spin determination by McClintock et al. (2006) is reliable. When the luminosity is high enough, the disk shadowing makes the measured spin lower than the actual value. This luminosity dependence of the measured spin is shown up both in our calculation and in the observational data. When the disk luminosity is medium (around 10^−0.2 Eddington luminosity), our results begin to diverge from the data. This may due to the photon scattering by the wind from the accretion disk as suggested by Done & Davis (2008). However, as our spectra fitting is based on the model of McClintock et al. (2006), so it is still possible that a different spectral model (e.g., a low temperature comptonization component; Middleton et al. 2006) may lead to a different result.

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