SPINS OF SUPERMASSIVE BLACK HOLES IN M87. II. FULLY GENERAL RELATIVISTIC CALCULATIONS

We assume that the accretion flows are axisymmetric, so the radiation fields have the same symmetry, and only the motions on the (R − θ) plane are required in our calculations. The general trajectories of photons in Kerr metric are described by three constants of motion (Bardeen et al. 1972). In terms of the photon's four-momentum, the conserved quantities are E = −p_t, the total energy at infinity; L = p_ϕ, the component of angular momentum parallel to symmetry axis; and $q=p_\theta ^2+\cos ^2\theta a^2 p_t^2+p_\phi ^2\cot ^2\theta$. By defining λ = p_ϕ/E and ${\cal Q}=q/E^2$, the four-momentum of photons can be rewritten as

$$\begin{equation} P_\mu =(p_t,p_r,p_\theta,p_\phi)=[-\!1,\pm \Delta ^{-1}\sqrt{{\cal R}(r)},\pm \sqrt{\Theta (\theta)}, \lambda]E, \end{equation} \tag{ 19 }$$

where

$$\begin{equation} {\cal R}(r)=r^4+(a^2-\lambda ^2-{\cal Q})r^2+2[{\cal Q}+(\lambda -a)^2]r-a^2{\cal Q}, \end{equation} \tag{ 20 }$$

$$\begin{equation} (\vartheta)={\cal Q}+a^2\cos ^2\theta -\lambda ^2\cot ^2\theta, \end{equation} \tag{ 21 }$$

where r = R/R_g.

Basically, the trajectory of a photon on the (R − θ) plane is governed by the geodesic equations

$$\begin{equation} {\mathscr{T}}= \pm \int _{r_0}^r\frac{{\rm d}r}{\sqrt{{\cal R}(r)}}=\pm \int _{\theta _0}^{\theta } \frac{{\rm d}\theta }{\sqrt{\Theta (\theta)}}, \end{equation} \tag{ 22 }$$

where ${\mathscr{T}}$ is the affine parameter and the ± signs represent the increment (+) or decrement (−) of r and θ coordinates along the trajectory. The reader is recommended to refer to Rauch & Blandford (1994), Cadez et al. (1998), and Li et al. (2005) for a comprehensive description of the analytic solutions of Equation (22). Given two constants of motion, λ and ${\cal{Q}}$, and the spin parameter a, the trajectory of a photon is uniquely determined. We neglect the photons with trajectories returning to the accretion disk but may encounter the TeV photons along their path because of rare probability of occurrence.

The specific flux density observed by a remote observer is expressed as

$$\begin{equation} F_{\nu _{\rm o}}=\int I_{\nu _{\rm o}}{\rm d}\Omega _{\rm o}, \end{equation} \tag{ 23 }$$

where $I_{\nu _{\rm o}}$ is the specific intensity as a function of frequency ν_o in the observer's frame and dΩ_o is the element of the solid angle subtended by the image of the accretion disk on the observer's sky. In a common way, the image of the disk can be described by two impact parameters α and β (see Figure 1), which respectively represent the displacement of the image perpendicular to the projection of the rotation of the black hole on the sky and the displacement parallel to the projection of the axis (Li et al. 2005). Applying the invariant I_ν/ν³ along the path of a photon (Rybicki & Lightman 1979, p. 146), we have

$$\begin{equation} F_{\nu _{\rm o}}=\int g^{-3}I_{\nu _{\rm e}}\frac{{\rm d}\alpha {\rm d}\beta }{D^2}, \end{equation} \tag{ 24 }$$

where ν_e is the frequency of the photons in the local rest frame (LRF) of the accretion flows, g = ν_o/ν_e is the redshift factor of the photons (see Section 3.3), $I_{\nu _{\rm e}}$ is the specific intensity of the disk radiation at radius r_e, and D is the distance of the black hole from the observer. Then the observed emergent spectrum is

$$\begin{equation} L_{\nu _{\rm o}}=4\pi D^2 F_{\nu _{\rm o}}=4\pi \int g^{-3}I_{\nu _{\rm e}}{\rm d}\alpha {\rm d}\beta. \end{equation} \tag{ 25 }$$

Zoom In Zoom Out Reset image size

For each (α, β) set, the constants of motion λ and $\cal Q$ can be expressed as (Li et al. 2005)

$$\begin{equation} \lambda =-\alpha \sin \Theta _{\rm obs},\quad {\cal Q}=\beta ^2+(\alpha ^2-a^2)\cos ^2\Theta _{\rm obs}, \end{equation} \tag{ 26 }$$

where the viewing angle Θ_obs is the inclination between the rotation axis of the accretion disk and the direction to the observer. Using the ray tracing method, we can locate the emission place r_e in the accretion disk for the photons which reach the observer's sky at point (α, β) and, therefore, obtain the radiation intensity $I_{\nu _{\rm e}}$ from the GR RIAF model. The observed emergent spectrum is obtained by integrating $I_{\nu _{\rm e}}$ over the observer's sky as shown in Equation (25).

Figure 1 shows the geometric scheme for calculations of the optical depth to TeV photons which emanate from the position P(R_TeV, Θ_TeV). The TeV photons unavoidably interact with the soft photons from the accretion disk when they are moving out (e.g., at the interacting point Q(r_c, θ_c)), setting up strong constraints on the radiation fields of soft photons. We delineate the interactions in the LNRF (see Figure 2). Firstly we divide the solid angle at the interacting point in the LNRF into numerous elements (Δθ_s, Δϕ_s). For each solid angle element at which the soft photons from the disk arrive, we can determine their constants of motion λ_s and ${\cal Q}_{\rm s}$ (see Equation (36)). Secondly we use the ray tracing method to trace the soft photons back to the accretion disk. We hence obtain the number density of soft photons. Lastly the optical depth is calculated by integrating all the soft photons from different directions along the trajectory of the TeV photons.

Zoom In Zoom Out Reset image size

Generally speaking, given the emission location of TeV photons (R_TeV, Θ_TeV) and the viewing angle of an observer (Θ_obs), there is a large number of trajectories to the observer at infinity because of the arbitrariness of the ϕ component. Since TeV photons along the shortest path intuitively suffer the least interactions and thus undergo the minimum optical depth, we only consider the optical depth for the shortest path in our calculations. We select the trajectory with α = 0 as the shortest path, which represents the case that the TeV photons lie at the projection of the rotation axis on the image sky. Keep in mind that genuine emission position always has nonzero ϕ-component, i.e., Φ_TeV ≠ 0, due to the drag of inertia caused by the rotation of black hole.

In the LNRF, given the frequency of TeV photons ν^TeV_∞ at infinity, we can determine their corresponding frequency at the interacting point Q(r_c, θ_c) by redshift factor defined as

$$\begin{equation} g_{_{\rm TeV}}=\frac{\nu _{\infty }^{\rm TeV}}{\nu _{r_c}^{\rm TeV}} =\frac{\left.e^\mu _{(t)}P_{\mu }^{\rm TeV}\right|_\infty }{\left.e^\mu _{(t)}P_{\mu }^{\rm TeV}\right|_{r_{\rm c}}} =\left.\frac{\Sigma ^{1/2}\Delta ^{1/2}}{A^{1/2}}\frac{1}{1-\omega \lambda _{\rm TeV}} \right|_{r_{\rm c}}, \end{equation} \tag{ 27 }$$

where the subscript r_c or ∞ means their values are calculated at position (r_c, θ_c) or infinity and similarly hereinafter. In the same way, at the interacting point the frequency of soft photons is given by the redshift factor in comparison with their frequency at the emanating place r_d in the accretion disk,

$$\begin{equation} g_{\rm s}=\frac{\left.e^\mu _{(t)}(\rm LNRF)P_\mu ^{\rm s}\right|_{r_{\rm c}}}{\left.e^\mu _{(t)}(\rm LRF)P_\mu ^{\rm s}\right|_{r_{\rm d}}} =\frac{e^{-\nu }(1-\omega \lambda _{\rm s})|_{r_c}}{\left.\gamma _r\gamma _\phi e^{-\nu }\left[1-\Omega \lambda _{\rm s}\mp \frac{\displaystyle \beta _r {\cal R}(r)^{1/2}}{\displaystyle \gamma _\phi A^{1/2}}\right] \right|_{r_{\rm d}}}. \end{equation} \tag{ 28 }$$

The directional angles of the TeV and the soft photons can be obtained from the projection of their four-momentum onto the spatial directions of the LNRF, respectively. The direction cosines ($\alpha _{_{\rm TeV}},\beta _{_{\rm TeV}},\gamma _{_{\rm TeV}}$) of the TeV photons in the LNRF are given by

$$\begin{equation} \gamma _{_{\rm TeV}} \equiv \cos \theta _{_{\rm TeV}} = \frac{\left.e^\mu _{(r)}P_{\mu }^{\rm TeV}\right|_{r_{\rm c}}}{-\left.e^\mu _{(t)}P_{\mu }^{\rm TeV}\right|_{r_{\rm c}}} =\frac{\pm {\cal R}(r)^{1/2}}{A^{1/2}(1-\omega \lambda _{\rm TeV})}, \end{equation} \tag{ 29 }$$

$$\begin{equation} \beta _{_{\rm TeV}} \equiv \sin \theta _{_{\rm TeV}}\cos \phi _{_{\rm TeV}} =\frac{\left.e^\mu _{(\theta)}P_{\mu }^{\rm TeV}\right|_{r_{\rm c}}}{-\left.e^\mu _{(t)}P_{\mu }^{\rm TeV}\right|_{r_{\rm c}}} =\frac{\pm \Theta (\theta)^{1/2}\Delta ^{1/2}}{A^{1/2}(1-\omega \lambda _{\rm TeV})}, \end{equation} \tag{ 30 }$$

$$\begin{equation} \alpha _{_{\rm TeV}} \equiv \sin \theta _{_{\rm TeV}}\sin \phi _{_{\rm TeV}} =\frac{\big.e^\mu _{(\phi)}P_{\mu }^{\rm TeV}\big|_{r_{\rm c}}}{-\left.e^\mu _{(t)}P_{\mu }^{\rm TeV}\right|_{r_{\rm c}}} =\frac{\lambda _{\rm TeV}}{\sin \theta _{\rm c}} \frac{\Sigma \Delta ^{1/2}}{A(1-\omega \lambda _{\rm TeV})}, \end{equation} \tag{ 31 }$$

and those of the soft photons are given by replacing the superscript (subscript) "TeV" with "s" in the above equations:

$$\begin{equation} \gamma _{\rm s} \equiv \cos \theta _{\rm s}=\frac{\left.e^\mu _{(r)}P_{\mu }^{\rm s}\right|_{r_{\rm c}}}{-\left.e^\mu _{(t)}P_{\mu }^{\rm s}\right|_{r_{\rm c}}}, \end{equation} \tag{ 32 }$$

$$\begin{equation} \beta _{\rm s} \equiv \sin \theta _{\rm s}\cos \phi _{\rm s} =\frac{\left.e^\mu _{(\theta)}P_{\mu }^{\rm s}\right|_{r_{\rm c}}}{-\left.e^\mu _{(t)}P_{\mu }^{\rm s}\right|_{r_{\rm c}}}, \end{equation} \tag{ 33 }$$

$$\begin{equation} \alpha _{\rm s} \equiv \sin \theta _{\rm s}\sin \phi _{\rm s} =\frac{\big.e^\mu _{(\phi)}P_{\mu }^{\rm s}\big|_{r_{\rm c}}}{-\left.e^\mu _{(t)}P_{\mu }^{\rm s}\right|_{r_{\rm c}}}. \end{equation} \tag{ 34 }$$

Note that α²_s + β²_s + γ²_s = 1, just giving α_s and β_s, we can obtain λ_s and ${\cal Q}_{\rm s}$ from Equations (33) and (34). For simplicity, we define two denotations

$$\begin{equation} \mathscr{A}=\frac{\alpha _{\rm s}\sin \theta _{\rm c} A}{\Sigma \Delta ^{1/2}},\quad \mathscr{B}=\frac{\beta _{\rm s}A^{1/2}(1-\omega \lambda _{\rm s})}{\Delta ^{1/2}}, \end{equation} \tag{ 35 }$$

then we can express λ_s and ${\cal Q}_{\rm s}$ by α_s and β_s as

$$\begin{equation} \lambda _{\rm s}=\frac{\mathscr{A}}{1+\omega \mathscr{A}},\quad {\cal Q}_{\rm s}=\mathscr{B}^2-(a\cos \theta _{\rm c})^2+(\lambda _{\rm s}\cot \theta _{\rm c})^2. \end{equation} \tag{ 36 }$$

Having obtained λ_s and ${\cal Q}_{\rm s}$, the ray tracing method is used to determine the emission location r_d of the soft photons in the accretion disk.

The number density of the soft photons at the interacting point (r_c, θ_c) is

$$\begin{equation} n_{\rm ph}(\theta _{\rm s},\phi _{\rm s},\nu _{\rm s},r_{\rm c},\theta _{\rm c})=\frac{I_{\nu _{\rm s}}}{ch\nu _{\rm s}} =g_s^3\frac{I_{\nu _{\rm d}}}{ch\nu _s}, \end{equation} \tag{ 37 }$$

where h is the Planck's constant. Here we have applied the invariant I_ν/ν³ along the path of a photon. Finally, the expression for the optical depth of the disk radiation fields to the TeV photons is written as

$$\begin{equation} \tau _{_{\rm TeV}}(R_{\rm TeV},\Theta _{\rm TeV})=\int\!\!\int\!\!\int\sigma _{\gamma \gamma }(\nu _{_{\rm TeV}},\nu _{\rm s},\bar{\mu }) \frac{I_{\nu _{\rm d}}}{ch\nu _{\rm s}}g_{\rm s}^3{\rm d}\Omega {\rm d}\nu _{\rm s} {\rm d}l, \end{equation} \tag{ 38 }$$

where ${\rm d}l=e^{\nu }\Sigma {\rm d}\mathscr{T}$ is the proper length differential with ${\rm d}\mathscr{ T}$ defined to be differential of the affine parameter $\mathscr{T}$ along the TeV photons' trajectory (see Equation (22)), dΩ = sin θ_sdθ_sdϕ_s is the solid angle element in the LNRF, and $\bar{\mu }$ is the cosine of angle between the TeV photons and the soft photons. The cross-section of the two colliding photons with energy $\epsilon _{_{\rm TeV}}=h\nu _{_{\rm TeV}}/m_ec^2$ and _s = hν_s/m_ec² is given by Gould & Schréder (1967):

$$\begin{equation} \sigma _{\gamma \gamma }=\frac{3\sigma _{\rm T}}{16}(1-v^2)\left[(3-v^4)\ln \left(\frac{1+v}{1-v}\right)-2 v(2-v^2)\right] \end{equation} \tag{ 39 }$$

for pair productions, where σ_T is the Thompson cross-section and v is the velocity of electrons and positrons at the center of momentum frame in units of c related to $\epsilon _{_{\rm TeV}}$ and _s through

$$\begin{equation} (1-v^2)=\frac{2}{(1-\bar{\mu })\epsilon _s\epsilon _{_{\rm TeV}}}. \end{equation} \tag{ 40 }$$

The cross section depends on the energies of the interacting photons and their colliding angle. If the two colliding photons run in parallel ($\bar{\mu }=1$), their interaction disappears. The 10 TeV photons mostly interact with soft photons of 0.05 eV for header-to-header collisions ($\bar{\mu }=-1$). As shown below, in specific calculations, the energy of soft colliding photons will be modified due to energy shift and bending of their trajectories caused by the GR effects.

It should be pointed out that the GR effects on $\tau _{_{\rm TeV}}$ occur through three factors: (1) changing the global structures of RIAFs, as a result, causing dependence of the radiation fields on the spin parameter; (2) modifying the observed spectrum from the RIAFs; and (3) bending the trajectories of TeV photons. For different spins of black holes, the global structures of RIAFs become significantly different starting ∼100R_g inward. Once considering the second influence, the observed spectrum is dependent on the viewing angles. The third influence is nonnegligible unless the trajectories of TeV photons closely approach the black hole (∼10R_g). These three factors jointly influence the final optical depth to TeV photons.

The black hole spins enhance the GR effects on the RIAF structures, the emergent spectrum, and the photon trajectories. Detailed calculations show that both the surface density and the temperature of accretion flows at fixed radius increase with the spins, consequently, making the radiation more efficient with the spins. In light of the drag of the frame, the trajectories of TeV photons will be elongated and the probabilities of pair productions are thus enhanced. These two factors together increase $\tau _{_{\rm TeV}}$ with the spins. However, the GR influence on the photon trajectories appears within a radius ∼10R_g and, therefore, is less important at relatively large radius.

Figure 3 shows the dependence of $\tau _{_{\rm TeV}}$ on the black hole spins for given parameters $\dot{m}=2.5\times 10^{-3}$ and Θ_obs = Θ_TeV = 30° at four emission radii of TeV photons. Obviously, $\tau _{_{\rm TeV}}$ increases with the spins, and its dependence on the spins is more tight at smaller radius because the GR effects become more significant. We can find that $\tau _{_{\rm TeV}}\ge 1$ for a>0.7 at R_TeV = 15R_g. On the other hand, the TeV photons from R_TeV = 6R_g cannot escape from the radiation fields of the GR RIAFs for all spin ranges since $\tau _{_{\rm TeV}}\gg 1$.

Zoom In Zoom Out Reset image size

In Figure 4, we show $\tau _{_{\rm TeV}}$ as a strong function of R_TeV, the radius of the location of TeV photons, by fixing $\dot{m}=2.5\times 10^{-3}$ and Θ_obs = Θ_TeV = 30°. The radiation fields from RIAFs become more and more intensive when R_TeV approaches the black hole, leading to a deep dependence of $\tau _{_{\rm TeV}}$ on R_TeV. It is difficult to get an analytical formulation of the dependence, but Figure 4 shows the details of the dependence numerically. The radiation fields from the RIAFs have different spatial distributions with the spins since the GR effects cause their structures distinct. As explained in Section 4.1, the spins enhance the radiation fields. If we fix the energy budget released from the RIAFs, the radiation fields should be more compact for the larger spins, causing a steeper $\tau _{_{\rm TeV}}$–R_TeV relation. The R_TeV dependence of $\tau _{_{\rm TeV}}$ can be applied to estimate the spins.

Zoom In Zoom Out Reset image size

The dependence of $\tau _{_{\rm TeV}}$ on Θ_obs and Θ_TeV is caused by the anisotropy of radiation fields from the disk and the dependence of the cross-section on the colliding angle in particular. Generally speaking, the number density of soft photons is mostly contributed from the inner region of the accretion disk. At larger R_TeV, the path of these photons that reach the interacting point with the TeV photons approximately has the same polar angle with Θ_TeV. Since when the two colliding photons run in parallel, their interaction disappears. We can see $\tau _{_{\rm TeV}}$ reaches its minimum at Θ_obs = Θ_TeV, whereas at smaller R_TeV, the TeV photons are embedded in the radiation fields from the inner region of the accretion disk and interact with soft photons from all directions and, consequently, it is difficult to give a general explanation to the dependence on Θ_obs and Θ_TeV.

We present the $\tau _{_{\rm TeV}}$-dependence on Θ_obs and Θ_TeV in Figures 5 and 6, respectively, for the fixed parameters $\dot{m}=6.3\times 10^{-3}$ and a = 0.998 at different R_TeV. From Figure 5, we find that (1) for Θ_TeV = 0°, $\tau _{_{\rm TeV}}$ monotonously increases with Θ_obs, and (2) there is a minimum $\tau _{_{\rm TeV}}$ for relatively large R_TeV at Θ_obs = Θ_TeV. This is confirmed by the case of Θ_TeV = 45°. Figure 6 shows that the $\tau _{_{\rm TeV}}$-dependence on Θ_TeV has the similar properties as that on Θ_obs.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The $\tau _{_{\rm TeV}}$-dependence on accretion rates is more straightforward in light of the changes of number density of soft photons and SED from the RIAFs. Figure 7 illustrates how $\tau _{_{\rm TeV}}$ changes with accretion rates for fixed parameters a = 0.998 and Θ_obs = Θ_TeV = 30°. We find that $\tau _{_{\rm TeV}}$ is extremely sensitive to accretion rates as $\tau _{_{\rm TeV}}\varpropto \dot{m}^{2.5\hbox{--}5.0}$, and the power index tends to be flat at larger R_TeV. This strongly indicates that TeV photons can be violently diluted by minor changes in accretion rates. In this sense, intensive γ-ray emission cannot be detected in variable sources. This has important implication in observations.

Zoom In Zoom Out Reset image size

This strong dependence can be explained in the following ways. Firstly, RIAFs have properties

$$\begin{equation} \Sigma _0\sim \dot{m}, B\sim \dot{m}^{1/2}, \nu _{\rm syn}\sim \gamma _e^2\dot{m}^{1/2}, \tau _{\rm es}\sim \dot{m}, \end{equation} \tag{ 41 }$$

where B is the magnetic field, ν_syn is peak frequency of synchrotron emission, γ_e = kT_e/m_ec² is the Lorentz factor of hot thermal electrons, and τ_es is the optical depth for Thompson scattering. The synchrotron emission power approximates as $\sim \gamma _e^2\Sigma _0 B^2\sim \gamma _e^2\dot{m}^2$ and the bremsstrahlung emission power $\sim \dot{m}^2$. In our calculations, we find that $T_e\propto \dot{m}^q$ and hence $\gamma _e\propto \dot{m}^q$, where q ∼ 1/3. To understand the $\tau _{_{\rm TeV}}$-dependence on $\dot{m}$, we have to know the energy of soft photons interacting with TeV photons. Figure 8 shows the contribution to $\tau _{_{\rm TeV}}$ for soft photons with different frequencies from an RIAF model with $\dot{m}=4.0\times 10^{-3}$, a = 0.998, Θ_obs = Θ_TeV = 30°, and R_TeV = 5R_g. We find that ν_s ∼ 10^14.5 Hz, which corresponds to the first Comptonization peak or the second determined by the parameters of the RIAFs. Since RIAFs are optically thin, most of the synchrotron photons escape and a small fraction (∼τ_es) are Compton-scattered by the hot electrons, the energy densities $U_{\rm C_1}\propto \gamma _e^2\dot{m}^3\propto \dot{m}^{3+2 q}$ and $U_{\rm C_2}\propto \gamma _e^2\dot{m}^4\propto \dot{m}^{4+2 q}$ for the first and second Comptonization, respectively. This determines the sensitivities of $\tau _{_{\rm TeV}}$-dependence on $\dot{m}$ as

$$\begin{equation} \tau _{_{\rm TeV}}\propto \left\lbrace\!\!\!\begin{array}{l}U_{\rm C_1}\propto \dot{m}^{3+2 q}\sim \dot{m}^{3.7},\\ U_{\rm C_2}\propto \dot{m}^{4+2 q}\sim \dot{m}^{4.7}.\end{array}\right. \end{equation} \tag{ 42 }$$

For larger $\dot{m}$, the bremsstrahlung emission will dominate the synchrotron emission and the Comptonization, even at the lower frequency. This leads to a flatter power index $\tau _{_{\rm TeV}}\varpropto \dot{m}^{2.5}$. These are nicely consistent with the numerical results as shown in Figure 7.

Zoom In Zoom Out Reset image size

We would like to point out that the very sensitive dependences of $\tau _{_{\rm TeV}}$ on accretion rate as $\tau _{_{\rm TeV}}\propto \dot{m}^{2.5\hbox{--}5.0}$ evidently indicate $\tau _{_{\rm TeV}}\gg 1$ for the standard accretion disk, immediately drawing a conclusion that the TeV photons cannot escape from the vicinity of SMBH fueled by the standard accretion disk. This is confirmed by the work of Zhang & Cheng (1997).

The radiation fields around the black hole can be quantified from the spectral fitting of M87. Before spectral fitting, it is useful to understand to what extent the different parameters influence on the structures of the RIAFs and hence their radiation fields. Small α_d indicates the low efficiency of angular momentum transfer, leading to the low radial velocity but high surface density; β_d represents the fraction of pressure contributed by the gas. As to lower β_d, the structures are changed through two ways: one by enhancing the magnetic field, and the other by increasing the temperature but reducing the surface density; $\dot{m}$ determines the overall peaks of the spectrum; the parameter δ, representing the fraction of the viscous dissipation that heats electrons, plays an important role in determining the temperature of the electrons and, therefore, in the energy boost of photons after Compton-scattering, i.e., δ mainly determines the frequency location of the Comptonization bumps. The value of δ is still open to question at present since some nonthermal mechanisms (e.g., Kolmgorov-like turbulent cascades and collisionless shocks) are highly unclear, which may significantly change the value of δ (Narayan & Yi 1995). We treat δ as a free parameter in our calculations. In the spectral fit, we set α_d and β_d a priori to reduce the freedom.

We use the GR RIAF model to fit the multiwavelength spectrum of M87. We firstly fit the observation data used in Paper I (hereafter Case I). We find that α_d = 0.025 and β_d = 0.2 give a good fit to the data. $\dot{m}$ and δ are adjusted with different spins. We furthermore take into account the HST observations (Sparks et al. 1996; Maoz et al. 2005) and redo the fit (hereafter Case II). We set the characteristic value of α_d = 0.1 and β_d = 0.5 which means equipartition between the gas and magnetized fields. We present the spectral fit in the left panels of Figures 9 and 10 for Case I and Case II, respectively. The fit parameters are listed in Table 2. The main differences between the two cases are the locations of the multi-Comptonization bumps. In terms of the energy dissipation to heat the electrons, the parameters α_d and δ can be absorbed into one parameter (see Equation (11)). We note that δα_d of Case II is larger by factor ∼4 to that of Case I, which leads to the difference by factor ∼4 on the location of the first Comptonization bump.

Table 2. Fit Parameters (Θ_obs = 30°)

Case	a	αd	βd	δ	$\dot{m}$	$R(\tau _{_{\rm TeV}}=1)/R_{\rm g}$
	0.0	0.025	0.20	0.39	2.0 × 10−4	28
Case I	0.8	0.025	0.20	0.18	1.8 × 10−4	20
	0.998	0.025	0.20	0.09	1.4 × 10−4	15
	0.0	0.1	0.5	0.35	1.0 × 10−4	28
Case II	0.8	0.1	0.5	0.18	9.0 × 10−5	20
	0.998	0.1	0.5	0.11	7.5 × 10−5	16

Download table as:  ASCIITypeset image

The accretion rates obtained from fitting of the SED in M87 can be examined independently. Di Matteo et al. (2003) give the Bondi accretion rate of the nucleus of M87 using the Chandra observation as an upper limit of the accretion rate, $\dot{M}<\dot{M}_{\rm Bondi}=1.6\times 10^{-3}\dot{M}_{\rm Edd}$. The accretion rate given by the RIAFs fitting in the current paper is consistent with this upper limit.

We list published literature that gives an estimate of jet power in M87 in Table 3. In regard to the inevitable uncertainties of the estimates, the jet power in M87 is in a range of 10⁴²–10⁴⁴ erg s ⁻¹. There are a number of theoretical calculations of jet power driven by BZ (Blandford–Znajek) or BP (Blandford–Payne) mechanisms. For example, Meier (2001) developed a hybrid jet model by combining the two processes, in which the jet power is given by

$$\begin{equation} \frac{L_{\rm jet}^{\rm Kerr-ADAF}}{L_{\rm acc}} =0.1\left(\alpha _{-1/2}^{\rm ADAF}\right)^{-1}(0.14 f^2+0.74 fj+j^2)g^2, \end{equation} \tag{ 43 }$$

where $f=\Omega _0/\Omega _{0,\rm NY}$, j = cJ/GM², $g=B_{\phi,0}/B_{\phi, \rm NY}$, and $L_{\rm acc}=0.1\dot{M}c^2$ (see Meier 2001 for details). Taking f = j = g ∼ 1 and α = 0.1, the jet power will be L_j ≈ 0.6L_acc. Indeed, if considering the field-enhancing shear by frame-dragging effects (Meier 2001; Nemmen et al. 2007), which are neglected in the self-similar solutions, we may have L_j>L_acc, corresponding to extracting the rotational energy from the black hole (the Penrose mechanism). The accretion rates obtained in our paper $\dot{m}\sim 10^{-4}$ are able to produce the jet power L_j>L_acc = 3 × 10⁴³ erg s⁻¹, which generally satisfies the jet energy budget from observations.

We calculate the optical depth to 10 TeV photons and show the results in the right panel of Figures 9 and 10, in which we set Θ_obs ∼ 30° according to the VLBI observation of the jets in M87 (Bicknell & Begelman 1996) since the jets generally align at the axis of the accretion disk. For both cases, the resultant $\tau _{_{\rm TeV}}$–R_TeV relation becomes steeper with the spins. If we define the transparent radius R_c as the radius at which $\tau _{_{\rm TeV}}=1$, we find R_c ≈ 15R_g and 28R_g for a = 0.998 and a = 0 in Case I, respectively, and R_c ≈ 16R_g and 28R_g in Case II. The variability of TeV emission at a timescale of ∼2 days constrains the emission region R_TeV ⩽ 20R_g (Aharonian et al. 2006; Paper I). We find that a black hole with a spin of $a_{_{\rm TeV}}=0.8$ for both cases leads to R_c = 20R_g. To avoid being optically thick to 10 TeV photons, it requires $\tau _{_{\rm TeV}}\lesssim 1$ at R_TeV = 20R_g. In this sense, the spin of SMBH in M87 should be a ⩾ 0.8 as shown in Figures 9 and 10.

We have to point out that the quality of the current data does not allow us to ascertain which fit is more practical to describe the accretion flows in M87. Future multiwavelength observations with high spatial resolution will provide stronger constraints on the spectrum and hence on the spin of the black hole in M87.