ENERGETIC GAMMA RADIATION FROM RAPIDLY ROTATING BLACK HOLES

When magnetized plasmas are accreting onto an astrophysical BH, the self-gravity of the plasma particles and the electromagnetic field little affects the spacetime geometry. Thus, around a rotating BH, the background geometry is described by the Kerr metric (Kerr 1963). In the Boyer–Lindquist coordinates, it becomes (Boyer & Lindquist 1967)

$$\begin{eqnarray}&&{{ds}}^{2}\;=\;{g}_{{tt}}{{dt}}^{2}+2{g}_{t\varphi }{dtd}\varphi +{g}_{\varphi \varphi }d{\varphi }^{2}+{g}_{{rr}}{{dr}}^{2}+{g}_{\theta \theta }d{\theta }^{2},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{g}_{{tt}}\equiv -\displaystyle \frac{{\rm{\Delta }}-{a}^{2}{\mathrm{sin}}^{2}\theta }{{\rm{\Sigma }}}{c}^{2},\qquad {g}_{t\varphi }\equiv -\displaystyle \frac{2({GM}/{c}^{2}){ar}{\mathrm{sin}}^{2}\theta }{{\rm{\Sigma }}}c,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{g}_{\varphi \varphi }\equiv \displaystyle \frac{A{\mathrm{sin}}^{2}\theta }{{\rm{\Sigma }}},\qquad {g}_{{rr}}\equiv \displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}},\qquad {g}_{\theta \theta }\equiv {\rm{\Sigma }};\end{eqnarray} \tag{ 4 }$$

${\rm{\Delta }}\equiv {r}^{2}-2({GM}/{c}^{2})r\quad +\quad {a}^{2}$, ${\rm{\Sigma }}\equiv {r}^{2}+{a}^{2}{\mathrm{cos}}^{2}\theta$, $A\equiv {({r}^{2}+{a}^{2})}^{2}-{\rm{\Delta }}{a}^{2}{\mathrm{sin}}^{2}\theta$, and $a\equiv J/({Mc})$, c denotes the speed of light, M the mass of the BH, J the BH's angular momentum. At the event horizon, Δ vanishes, giving ${r}_{{\rm{H}}}\equiv {r}_{{\rm{g}}}+\sqrt{{r}_{{\rm{g}}}{}^{2}-{a}^{2}}$ as the horizon radius, where ${r}_{{\rm{g}}}\equiv {{GMc}}^{-2}$.

In the same manner as in pulsar magnetospheres, the Gauss's law, ${{\rm{\nabla }}}_{\mu }{F}^{t\mu }\;=\;(4\pi /c)\rho$, gives the Poisson equation for the non-corotational potential Ψ (Hirotani 2006),

$$\begin{eqnarray}&&-\displaystyle \frac{{c}^{2}}{\sqrt{-g}}{\partial }_{\mu }\left(\displaystyle \frac{\sqrt{-g}}{{\rho }_{{\rm{w}}}^{2}}{g}^{\mu \nu }{g}_{\varphi \varphi }{\partial }_{\nu }{\rm{\Psi }}\right)\;=\;4\pi (\rho -{\rho }_{\mathrm{GJ}})\end{eqnarray} \tag{ 5 }$$

where the Greek indices run over t, r, θ, φ, $\sqrt{-g}\;=\;\sqrt{{g}_{{rr}}{g}_{\theta \theta }{\rho }_{{\rm{w}}}^{2}}\;=\;c{\rm{\Sigma }}\mathrm{sin}\theta$, ${\rho }_{{\rm{w}}}^{2}\equiv {g}_{t\varphi }^{2}-{g}_{{tt}}{g}_{\varphi \varphi }$, and ρ the real charge density; the general-relativistic (GR) Goldreich–Julian (GJ) charge density is defined as (Goldreich & Julian 1969; Mestel 1971; Hirotani 2006)

$$\begin{eqnarray}&&{\rho }_{\mathrm{GJ}}\equiv \displaystyle \frac{{c}^{2}}{4\pi \sqrt{-g}}{\partial }_{\mu }\left[\displaystyle \frac{\sqrt{-g}}{{\rho }_{{\rm{w}}}^{2}}{g}^{\mu \nu }{g}_{\varphi \varphi }({{\rm{\Omega }}}_{{\rm{F}}}-\omega ){F}_{\varphi \nu }\right].\end{eqnarray} \tag{ 6 }$$

The magnetic-field-aligned electric field can be computed by ${E}_{\parallel }\equiv ({\boldsymbol{B}}/B)$ $\cdot {\boldsymbol{E}}\;=\;({\boldsymbol{B}}/B)\cdot (-{\rm{\nabla }}{\rm{\Psi }}).$

If ρ deviates from ${\rho }_{\mathrm{GJ}}$ in any region, ${E}_{\parallel }$ is exerted there along ${\boldsymbol{B}}$. Once ${E}_{\parallel }$ arises, positive and negative charges migrate in opposite directions. For the magnetosphere to be force-free outside the gap, ρ should match ${\rho }_{\mathrm{GJ}}$ at the boundaries. Since ρ has opposite signs at the two boundaries, ${E}_{\parallel }$ should arise around the null-charge surface, where ${\rho }_{\mathrm{GJ}}$ vanishes, in the same way as pulsars (Romani 1996; Cheng et al. 2000). On these grounds, we consider that the null surface is a natural place for a particle accelerator to arise.

In Figure 1, we plot the distribution of the null surface as the thick red solid curve. In the left panel, we assume that the magnetic field is split-monopole, adopting ${A}_{\varphi }\propto -\mathrm{cos}\theta$ as the magnetic flux function (Michel 1973). In the right panel, on the other hand, we assume a parabolic magnetic field line with ${A}_{\varphi }\propto r(1-\mathrm{cos}\theta )$ on the poloidal plane (i.e., r–θ plane). We adopt ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;0.3{\omega }_{{\rm{H}}}$ for both cases (McKinney et al. 2012; Beskin & Zheltoukhov 2013). It follows that the null surface distributes nearly spherically, irrespective of the poloidal magnetic field configuration. This is because its position is essentially determined by the condition $\omega \;=\;{{\rm{\Omega }}}_{{\rm{F}}}({A}_{\varphi })$ because ω has weak dependence on θ near the horizon, and because we assume ${{\rm{\Omega }}}_{{\rm{F}}}$ is constant for A_φ for simplicity. If ${{\rm{\Omega }}}_{{\rm{F}}}$ decreases (or increases) toward the polar region, the null surface shape becomes prolate (or oblate).

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The field angular frequency ${{\rm{\Omega }}}_{{\rm{F}}}$ is, indeed, deeply related to the accretion conditions. For instance, in order to get adequate jet efficiency from the magnetic field in the funnel above the BH for a radio-loud active galactic nuclei (AGNs), one needs significant flux compression from the lateral boundary imposed by the accretion flow and corona (McKinney et al. 2012; Sikora & Begelman 2013). These lateral boundary conditions and the plasma injection process will modify ${{\rm{\Omega }}}_{{\rm{F}}}$ in the funnel to $-0.10{\omega }_{{\rm{H}}}\lt {{\rm{\Omega }}}_{{\rm{F}}}\lt 0.35{\omega }_{{\rm{H}}}$ (McKinney et al. 2012; Beskin & Zheltoukhov 2013). For illustrative purposes, we simplify this behavior by choosing a constant ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;0.3{\omega }_{{\rm{H}}}$ for A_φ. We acknowledge that this is not a self consistent solution, but we believe that it is sufficient for demonstrating the physical process that we propose. In Section 5.9, we consider the cases of different ${{\rm{\Omega }}}_{{\rm{F}}}$ and compare the results.

In addition to the null surface discussed above, a stagnation surface is also considered to be a plausible place for particle accelerators to arise. Since both inflows and outflows start from the stagnation surface in magnetohydrodynamics (MHD), we can expect that the plasma density becomes low, thereby leading to a non-vanishing ${E}_{\parallel }$ there. Levinson & Rieger (2011, LR11) and Broderick & Tchekhovskoy (2011, BT15) examined this possibility. In this subsection, we examine only the position of the stagnation surface, leaving its electrodynamical plausibility as a particle accelerator site to Section 2.3.

In a stationary and axisymmetric BH magnetosphere, MHD inflows and outflows start from the two-dimensional surface on which

$$\begin{eqnarray}&&{k}_{0}^{\prime} \;=\;0\end{eqnarray} \tag{ 7 }$$

holds (Takahashi et al. 1990), where

$$\begin{eqnarray}&&{k}_{0}\equiv -{g}_{{tt}}-2{g}_{t\varphi }{{\rm{\Omega }}}_{{\rm{F}}}-{g}_{\varphi \varphi }{{\rm{\Omega }}}_{{\rm{F}}}{}^{2},\end{eqnarray} \tag{ 8 }$$

and the prime denotes the derivative along the poloidal magnetic field line. The condition (7) is equivalent with imposing a balance among the gravitational, centrifugal, and Lorentz forces on the poloidal plane. In Figure 2, we plot the contours of k₀ for split-monopole and parabolic poloidal magnetic field lines. Far away from the horizon, we obtain ${k}_{0}\approx {c}^{2}-{\varpi }^{2}{{\rm{\Omega }}}_{{\rm{F}}}{}^{2}$, where ϖ denotes the distance from the rotation axis. Thus, at large ϖ, the potential k₀ becomes negative, as denoted by the red dotted contours on the right part of each panel. Note that k₀ vanishes at $\varpi \approx c/{{\rm{\Omega }}}_{{\rm{F}}}$. This two-dimensional surface is called the outer light surface and depicted by the thick red solid curve in Figure 2. If the general-relativistic effects (i.e., redshift and frame-dragging effects) were negligible, and if ${{\rm{\Omega }}}_{{\rm{F}}}$ is constant for the magnetic flux function A_φ, the outer light surface would distribute cylindrically, as in pulsar magnetospheres. Outside the outer light surface, plasma particles must flow outwards, if they are frozen-in to the magnetic field. Inside the outer light surface, k₀ becomes positive, as denoted by the dashed contours. Further inside, close to the horizon, k₀ begins to decrease again due to the strong gravity of the hole. As a result, another light surface, where k₀ vanishes, appears, as depicted by the thick red solid curve near the horizon. Inside this inner light surface, plasma particles must flow inward if they are frozen-in to the magnetic field.

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As demonstrated by Takahashi et al. (1990), a stationary and axisymmetric MHD flow flows from a greater k₀ region to a smaller k₀ one. It follows that both the inflows and outflows start from the two-dimensional surface on which k₀ maximizes along the poloidal magnetic field line. Thus, putting ${k}_{0}^{\prime} \;=\;0$, we obtain the position of the stagnation surface. In Figure 2, we plot the stagnation surface as the thick green solid curve. It is clear that the stagnation surface is located at $r\lt 5{r}_{{\rm{g}}}$ in the lower latitudes but at $r\gt 5{r}_{{\rm{g}}}$ it is in the higher latitudes, irrespective of the magnetic field-line configuration, as long as ${{\rm{\Omega }}}_{{\rm{F}}}$ is a good fraction of ${\omega }_{{\rm{H}}}$, e.g., $0.25{\omega }_{{\rm{H}}}\lt {\rm{\Omega }}\lt 0.50{\omega }_{{\rm{H}}}$. It is, however, noteworthy that the numerical simulations (McKinney et al. 2012) and a supporting analytical work (Beskin & Zheltoukhov 2013) claim $-0.10{\omega }_{{\rm{H}}}\lt {\rm{\Omega }}\lt 0.35{\omega }_{{\rm{H}}}$ in the funnel to represent a radio-loud AGN. As the poloidal field geometry changes from radial to parabolic, the stagnation surface moderately moves away from the rotation axis in the higher latitudes.

This analytical result was confirmed by GR MHD simulations (McKinney 2006; Broderick & Tchekhovskoy 2015). In these numerical works, the stagnation surface is time dependent but stably located at 5–10 ${r}_{{\rm{g}}}$ with a prolate shape, as depicted in Figure 2. In addition, the position of the stagnation surface changes as a function of the accretion rate and other numerical settings such as the initial and boundary conditions. However, it should be emphasized that these changes are caused merely through the poloidal magnetic field configuration and ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;{{\rm{\Omega }}}_{{\rm{F}}}({A}_{\varphi })$, the latter of which defines the potential k₀.

To examine the Poisson Equation (5), it is worth noting that the distribution of ${\rho }_{\mathrm{GJ}}$ has relatively small dependence on the magnetic field geometry near the horizon because its distribution is essentially governed by the spacetime dragging effect, as indicated by the red solid, dashed, and dotted contours in Figure 1 for split-monopole and parabolic fields. As a result, the gap solution little depends on the magnetic field-line configuration, particularly in the higher latitudes, forming a striking contrast to pulsar outer-gap (OG) models, in which the field-line configuration defines ${\rho }_{\mathrm{GJ}}$.

As for ${k}_{0}\;=\;{k}_{0}(r,\theta )$, it has no dependence on the field geometry, as long as ${{\rm{\Omega }}}_{{\rm{F}}}$ is constant for all the field lines. In this particular case, the stagnation surface distributes more or less similarly for split-monopole and parabolic fields, as Figure 2 indicates. Thus, in what follows, we assume a split-monopole field and adopt a constant ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;{{\rm{\Omega }}}_{{\rm{F}}}({A}_{\varphi })$ for simplicity, as described in the left panels of Figures 1 and 2. However, it is noteworthy that the lateral boundary conditions and the plasma injection in the tenuous funnel alter both the magnetic-field distribution near the event horizon and ${{\rm{\Omega }}}_{{\rm{F}}}$ in an actual solution (Phinney 1983; McKinney et al. 2012; Beskin & Zheltoukhov 2013).

Let us leave the geometrical argument of the stagnation surface and turn to the electrodynamics of a gap, which may be formed around the null surface or the stagnation surface. For this sake, we must consider the Poisson Equation (5) for the non-corotational potential Ψ.

If the full gap width w along the magnetic field line is short compared to the meridional and azimuthal dimensions of the gap, only the derivatives along the poloidal magnetic field line remains in Equation (5). Noting that the gravitational forces are negligible compared to the electrodynamical forces in the gap (except for the intimate vicinity of the horizon), we can reduce Equation (5) into

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\parallel }}{{ds}}\;=\;4\pi (\rho -{\rho }_{\mathrm{GJ}}),\end{eqnarray} \tag{ 9 }$$

where s denotes the distance along the poloidal magnetic field line measured outward in the Boyer–Lindquist coordinates. We define s = 0 to indicate the null surface, $r\;=\;{r}_{0}$. Thus, the position s that satisfies ${r}_{0}+s\;=\;{r}_{{\rm{H}}}$ represents the event horizon because we adopt radial magnetic field lines on the poloidal plane. Both the inner boundary (at $s\;=\;{s}_{1}$) and the outer boundary (at $s\;=\;{s}_{2}$) are free boundaries and are determined as a function of the accretion rate by the procedure to be described in Section 4.5. If the real charge density ρ is greater (or smaller) than ${\rho }_{\mathrm{GJ}}$, ${E}_{\parallel }$ increases (or decreases) outward.

The created ${e}^{\pm }$ pairs are separated by ${E}_{\parallel }$ in the gap. Without loss of any generality, we can assume ${E}_{\parallel }\lt 0$ in the gap, which is possible if ${\boldsymbol{B}}\cdot {\boldsymbol{L}}\gt 0$ (or exactly speaking, if ${A}_{\varphi ,\theta }\gt 0$) in the northern hemisphere near the horizon, where ${\boldsymbol{L}}$ denotes the BH's angular momentum vector. Note that the sign of ${E}_{\parallel }$ changes from pulsar outer gaps because the null surface is formed by the frame dragging instead of the convex topology of the poloidal magnetic field lines. The negative ${E}_{\parallel }$ results in an outwardly decreasing dimensionless charge density, $\rho /({\rm{\Omega }}B/2\pi c)$. Since the total current per magnetic flux tube conserves, the sum of positronic and electronic charge densities per magnetic flux tube also conserves. Thus, pair production per unit length, $d(\rho /B)/{ds}$, becomes more or less constant for s, provided that the background soft-photon density is roughly homogeneous, as demonstrated in Figure 5 of Hirotani & Shibata (1999). We therefore assume that $\rho /B$ decreases linearly with s in the present paper. In general, to analyze the spatial dependence of $\rho /B$, we must solve the Boltzmann equations of electrons, positrons, and photons simultaneously together with the Poisson Equation (9), as Hirotani & Shibata (1999) did for pulsar magnetospheres.

In the present paper, we employ a simplified model such that $\rho /B$ decreases with increasing s linearly as

$$\begin{eqnarray}&&\displaystyle \frac{\rho (s)}{B(s)}\;=\;j\displaystyle \frac{{\rho }_{\mathrm{GJ}}({s}_{2})}{B({s}_{2})}\displaystyle \frac{2\;s-{s}_{1}-{s}_{2}}{{s}_{2}-{s}_{1}},\end{eqnarray} \tag{ 10 }$$

where $0\lt j\leqslant 1$. Note that $s\;=\;{s}_{2}$ denotes the outer boundary and that $\rho ({s}_{2})$ is negative. If we put j = 0, we obtain a vacuum gap. In a non-vacuum gap, $j\ne 0$, ${E}_{\parallel }$ is partially screened by the created and polarized pairs. If j = 1, we obtain $\rho ({s}_{2})\;=\;{\rho }_{\mathrm{GJ}}({s}_{2})$. A super-GJ current, $j\gt 1$, is prohibited, as will be discussed in Section 6.3.2.

2.3.1. Vacuum Gap around the Null Surface

Let us begin with the vacuum case, j = 0, which leads to $\rho \;=\;0$. The Poisson Equation (9) shows that ${{dE}}_{\parallel }/{ds}\gt 0$ (or ${{dE}}_{\parallel }/{ds}\lt 0$) holds in the outer (or inner) part of the gap because $\rho -{\rho }_{\mathrm{GJ}}\;=\;-{\rho }_{\mathrm{GJ}}$ becomes positive (or negative) in the outer (or inner) part of the gap. Thus, we obtain ${E}_{\parallel }\lt 0$ in the gap, and find that $| {E}_{\parallel }|$ maximizes at the point where ρ matches ${\rho }_{\mathrm{GJ}}$, which is realized at the null surface, s = 0, in a vacuum gap. Outside the gap, on the other hand, to meet the force-free condition, ρ should match ${\rho }_{\mathrm{GJ}}$. It results in a jump of ${{dE}}_{\parallel }/{ds}$ at the boundaries, which should be compensated by the surface charge there.

For analytical purpose, in this particular subsection (Section 2.3.1), we assume that the full gap width, w, is much smaller compared to ${r}_{{\rm{g}}}$, and expand Equation (9) around the null surface (s = 0) to obtain

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{\parallel }}{{ds}}\;=\;-4\pi {\rho }_{\mathrm{GJ}}^{\prime} s,\end{eqnarray} \tag{ 11 }$$

where ${\rho }_{\mathrm{GJ}}^{\prime}$ denotes the derivative of ${\rho }_{\mathrm{GJ}}$ at s = 0. Imposing ${E}_{\parallel }\;=\;0$ at both boundaries ($s\;=\;\pm w/2$), we obtain

$$\begin{eqnarray}&&{E}_{\parallel }(s)\;=\;2\pi {\rho }_{\mathrm{GJ}}^{\prime} [{(w/2)}^{2}-{s}^{2}].\end{eqnarray} \tag{ 12 }$$

The distribution of $\rho /B$, ${\rho }_{\mathrm{GJ}}/B$, and ${E}_{\parallel }$ are illustrated in Figure 3. As the gap width w increases, $| {E}_{\parallel }|$ increases quadratically. Integrating Equation (12) over s from ${s}_{1}\;=\;-w/2$ to ${s}_{2}\;=\;+w/2$, we obtain the electric potential drop in the gap,

$$\begin{eqnarray}&&{V}_{\mathrm{gap}}\;=\;\displaystyle \frac{8}{3}\pi {\rho }_{\mathrm{GJ}}^{\prime} {(w/2)}^{3}\approx {\left(\displaystyle \frac{w/2}{{r}_{{\rm{H}}}}\right)}^{3}(\mathrm{EMF}),\end{eqnarray} \tag{ 13 }$$

where the electromotive force exerted on the hemisphere of the horizon is given by

$$\begin{eqnarray}&&\mathrm{EMF}\equiv \displaystyle \frac{{{\rm{\Omega }}}_{{\rm{F}}}}{{\rm{c}}}{{\rm{r}}}_{{\rm{H}}}{}^{2}{\rm{B}},\end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray}&&{\rho }_{\mathrm{GJ}}^{\prime} \approx \displaystyle \frac{{{\rm{\Omega }}}_{{\rm{F}}}B}{2\pi c}\displaystyle \frac{1}{{r}_{{\rm{H}}}}\end{eqnarray} \tag{ 15 }$$

is used. Although we expand ${\rho }_{\mathrm{GJ}}$ around s = 0, the argument on ${V}_{\mathrm{gap}}$ is basically valid even when $w\gt {r}_{{\rm{g}}}$ because ${\rho }_{\mathrm{GJ}}$ changes from 0 (at the null surface) to $\approx ({\omega }_{{\rm{H}}}-{{\rm{\Omega }}}_{{\rm{F}}})B/(2\pi c)$ (at the horizon) in any case.

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If we assume some w, we can solve ${E}_{\parallel }(s)$ with Equation (9). For a vacuum gap, there is no photon flux that is needed to realize the pair-creation cascade that closes the gap. Thus, we cannot solve w for a given accretion rate. Thus, we can only illustrate by hand how $\rho (s)$ and ${E}_{\parallel }(s)$ are related in Figure 3. A vacuum gap can be naturally formed around the null surface because ${{dE}}_{\parallel }/{ds}$ changes sign at the null surface. There remains, however, another problem: how to supply the charges that emit photons and how to realize a force-free magnetosphere outside the gap. This motivates us to consider pair creation inside and outside the gap.

2.3.2. Non-Vacuum Gap around the Null Surface

To pursue the supply of charges into the magnetosphere, we next consider a non-vacuum gap, $0\lt j\leqslant 1$. In a non-vacuum gap, ${\rho }_{\mathrm{GJ}}$ is partly canceled by ρ, leading to a smaller $| {E}_{\parallel }|$ than the vacuum value. To grasp the relationship between $\rho (s)$ and ${E}_{\parallel }(s)$, we only show the representative results of their distribution in this section, leaving the further details of electrodynamics in Section 5.

We solve the gap by the method described in Section 5 for the BH magnetosphere of IC 310, adopting j = 0.5 and and dimensionless accretion rate $\dot{m}\;=\;3.16\times {10}^{-5}$, where

$$\begin{eqnarray}&&\dot{m}\equiv \displaystyle \frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}};\end{eqnarray} \tag{ 16 }$$

$\dot{M}$ denotes the mass accretion rate, ${\dot{M}}_{\mathrm{Edd}}$ the Eddington accretion rate

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{Edd}}\equiv \displaystyle \frac{{L}_{\mathrm{Edd}}}{{\eta }_{\mathrm{eff}}{c}^{2}}\;=\;1.39\times {10}^{27}{M}_{9}{\rm{g}}\;{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 17 }$$

${L}_{\mathrm{Edd}}$ the Eddington luminosity, and the conversion efficiency is assumed to be ${\eta }_{\mathrm{eff}}\;=\;0.1$. The resultant $\rho (s)$ and ${E}_{\parallel }(s)$ distribution is presented in Figure 4, The gap inner and outer boundaries are located at ${s}_{1}\;=\;-0.201{r}_{{\rm{g}}}$ and ${s}_{2}\;=\;2.345{r}_{{\rm{g}}}$, respectively. Quantities $\rho /B$, ${\rho }_{\mathrm{GJ}}/B$, and ${E}_{\parallel }$ are depicted only within the gap, ${s}_{1}\lt s\lt {s}_{2}$. The maximum ${E}_{\parallel }$ becomes $6.19\times {10}^{1}\;\mathrm{statvolt}\;{\mathrm{cm}}^{-1}$. Because of the screening by the discharged pairs, this value is about half of the vacuum-case (i.e., j = 0 case) value ${E}_{\parallel }(0)\;=\;1.35\quad \times \quad {10}^{2}\;{\mathrm{statvoltcm}}^{-1}$ for the same $w/2\;=\;{s}_{2}$ (Equation (12)). The potential drop becomes ${V}_{\mathrm{gap}}\;=\;4.34\times {10}^{14}$ statvolt for the j = 0.5 case.

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We also present the case of the marginally super-GJ case, j = 1, in Figure 5. Since $\rho \;=\;{\rho }_{\mathrm{GJ}}$ holes at the outer boundary, ${{dE}}_{\parallel }/{ds}$ vanishes there, requiring no surface charge. The maximum ${E}_{\parallel }$ becomes $2.08\times {10}^{1}\;\mathrm{statvolt}\;{\mathrm{cm}}^{-1}$, and the potential drop becomes ${V}_{\mathrm{gap}}\;=\;1.72\times {10}^{14}$ V. Note that the total luminosity of the gap is given by the product of ${V}_{\mathrm{gap}}$ and the current flowing through the gap, the latter of which is proportional to j. Since the product ${{jV}}_{\mathrm{gap}}$ decreases only 20% from j = 0.5 to j = 1, we find that the gap luminosity depends on j relatively weakly, as long as $0.5\leqslant j\leqslant 1$. We thus adopt j = 1 as the representative value in the present paper.

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Figures 4 and 5 show that the gap width $w\;=\;{s}_{2}-{s}_{1}$ slightly increases from j = 0.5 to j = 1.0. This is because the weaker ${E}_{\parallel }$ for j = 1 results in less energetic IC photons, which require longer mean-free path to materialize as pairs, thereby increasing w. However, at the same time, the doubled particle flux of the j = 1 case (from the j = 0.5 case) results in an increased IC photon flux, which in turn contributes to a decreased w (for details, see Hirotani 2013). Because of such negative feedback effects, the gap solution exists in a vast expanse of parameter space in the BH and pulsar magnetospheres. For this reason, the classic pulsar vacuum OG model works qualitatively well because the errors incurred by the vacuum assumption and the GJ charge density in the gap partly cancel each other and keep the conclusions more or less close to the non-vacuum-gap model.

The maximum current density flowing through the gap is limited by $c| {\rho }_{\mathrm{GJ}}(s\;=\;{s}_{2})|$. Thus, the wider the gap extends, the greater $c| {\rho }_{\mathrm{GJ}}({s}_{2})|$ becomes. For example, near the null surface, we can expand ${\rho }_{\mathrm{GJ}}$ to obtain

$$\begin{eqnarray}&&c| {\rho }_{\mathrm{GJ}}({s}_{2})| \approx c| {\rho }_{\mathrm{GJ}}^{\prime} | {s}_{2}.\end{eqnarray} \tag{ 18 }$$

Integrating over the hemisphere of the horizon, we obtain the total current flowing in the entire magnetosphere,

$$\begin{eqnarray}&&{J}_{\mathrm{max}}\approx (c| {\rho }_{\mathrm{GJ}}^{\prime} | {s}_{2})\cdot 2\pi {r}_{{\rm{H}}}{}^{2}\approx {J}_{\mathrm{tot}}\displaystyle \frac{{s}_{2}}{{r}_{{\rm{H}}}},\end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray}&&{J}_{\mathrm{tot}}\equiv \displaystyle \frac{({\omega }_{{\rm{H}}}-{{\rm{\Omega }}}_{{\rm{F}}})B}{2\pi }\cdot 2\pi {r}_{{\rm{H}}}{}^{2}\;=\;({\omega }_{{\rm{H}}}-{{\rm{\Omega }}}_{{\rm{F}}}){{Br}}_{{\rm{H}}}{}^{2}\end{eqnarray} \tag{ 20 }$$

represents the maximally possible electric current flowing along the magnetic field lines threading the horizon. The ${\omega }_{{\rm{H}}}-{{\rm{\Omega }}}_{{\rm{F}}}$ factor in ${J}_{\mathrm{tot}}$ comes from the fact that ${\rho }_{\mathrm{GJ}}$ is proportional to ${\omega }_{{\rm{H}}}-{{\rm{\Omega }}}_{{\rm{F}}}$ at the horizon.

Combining Equations (13) and (19), we obtain the maximally possible luminosity of the gap,

$$\begin{eqnarray}&&{L}_{\mathrm{gap},\mathrm{max}}\;=\;{V}_{\mathrm{gap}}{J}_{\mathrm{max}}\approx (\mathrm{EMF})\cdot {J}_{\mathrm{tot}}{\left(\displaystyle \frac{{s}_{2}}{{r}_{{\rm{H}}}}\right)}^{4}\;=\;{L}_{\mathrm{sd}}{\left(\displaystyle \frac{{s}_{2}}{{r}_{{\rm{H}}}}\right)}^{4}.\end{eqnarray} \tag{ 21 }$$

If ${E}_{\parallel }$ took the vacuum value (i.e., Equation (12)) but the current was ${J}_{\mathrm{max}}$, which would contradict each other, the gap luminosity would become ${L}_{\mathrm{gap},\mathrm{max}}$. In a non-vacuum gap, ${E}_{\parallel }$ is less than the vacuum value and the current is less than ${J}_{\mathrm{max}};$ thus, the gap luminosity should be less than Equation (21). The ${s}_{2}{}^{4}\approx {(w/2)}^{4}$ dependence comes from the fact that ${E}_{\parallel }$ is proportional to ${s}_{2}{}^{2}$, that the potential drop is proportional to ${E}_{\parallel }{s}_{2}$, and that the maximum current is also proportional to s₂ (Equation (19)).

In short, a non-vacuum gap can be formed around the null surface because $\rho -{\rho }_{\mathrm{GJ}}$, and hence ${{dE}}_{\parallel }/{ds}$ naturally changes sign within the gap, as demonstrated in Figures 4 and 5.

2.3.3. Gap Formation Away from the Null Surface

Let us next examine the possibility of the formation of a gap away from the null surface, e.g., around the stagnation surface. In Figure 6, we sketch the possibility of the formation of a gap around $s\approx 1.57{r}_{{\rm{g}}}$. If there is no particle injection across the boundaries, only electrons exist at the outer boundary, and only positrons at the inner boundary (top panel). In this case, $\rho -{\rho }_{\mathrm{GJ}}$ is positive definite (middle panel), resulting in a monotonically increasing ${E}_{\parallel }$ in the gap (bottom panel). Thus, the inner boundary cannot be formed in this case if we set ${E}_{\parallel }\;=\;0$ at the outer boundary.

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To change the sign of $\rho -{\rho }_{\mathrm{GJ}}$, we must consider the injection of electrons across the inner boundary, in the same way as in a pulsar outer-gap model (Hirotani & Shibata 2001a). It is possible if a small-amplitude, residual ${E}_{\parallel }$ separates the charges at $s\lt {s}_{1}$, and if a fraction of the returned electrons enters the gap. In this case, the position of the gap center is determined solely by the injected current across the boundary. Thus, if the injected current density per magnetic flux tube, ${ce}{({n}_{-}/B)}_{1}$, coincidentally matches $c{\rho }_{\mathrm{GJ}}/B$ at the stagnation surface, the gap appears around the stagnation surface, where the subscript 1 shows that the quantity is evaluated at the inner boundary. Because $\rho -{\rho }_{\mathrm{GJ}}$ changes sign (middle panel of Figure 7), ${E}_{\parallel }$ also changes sign (bottom panel) to close the gap. We interpret the treatment of LR11 and BT15 to correspond to this case, although it is not explicitly mentioned in their papers.

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In short, the position of a gap shifts outward (or inward) if there is a leptonic current injection across the inner (or outer) boundary. The center of the gap is determined by the strength of the injected current. Only when the injected current density ${ce}({n}_{-}/B)$ at the inner boundary matches $c{\rho }_{\mathrm{GJ}}/B$ at the stagnation surface can the gap center be located at the stagnation surface.

In BH gap models, there have been two main ideas about the position of the gap. One considers gaps at the null surface (Beskin et al. 1992, HO98), and the other at the separation surface (LR11, BT15). In this paper, we pursue the first possibility because there is no essential difference in gap electrodynamics between the two cases and because no current injection may be the simplest assumption. For instance, the gap closure condition (Section 4.5) is much simplified in the former case because we do not have to consider additional γ-ray emission by the injected charges.

If the curvature drag force balances ${{eE}}_{\parallel }$, the electron Lorentz factors are saturated at the curvature-limited value,

$$\begin{eqnarray}&&{\gamma }_{\mathrm{curv}}\equiv {\left(\displaystyle \frac{3{\rho }_{{\rm{c}}}^{2}}{2e}{E}_{\parallel }\right)}^{1/4},\end{eqnarray} \tag{ 30 }$$

where ${\rho }_{{\rm{c}}}$ denotes the curvature radius of the three-dimensional particle path, which becomes comparable to the curvature radius of the magnetic field lines in a rotating magnetosphere (Hirotani 2011). Although the split-monopole solution specifies not only the poloidal magnetic field components but also the toroidal (i.e., azimuthal) one, we set ${\rho }_{{\rm{c}}}$ as a free parameter in the present paper because a time-dependence accretion onto a BH will have a spatially and temporary varying magnetic field near the horizon (Krolik et al. 2005).

If the IC drag force balances ${{eE}}_{\parallel }$, on the other hand, the saturated Lorentz factor, ${\gamma }_{\mathrm{IC}}$, should be implicitly solved from the balance

$$\begin{eqnarray}&&\displaystyle \frac{1}{c}{\displaystyle \int }_{{E}_{\mathrm{min}}}^{{\gamma }_{\mathrm{IC}}{m}_{{\rm{e}}}{c}^{2}}{E}_{\gamma }{{dE}}_{\gamma }{\displaystyle \int }_{{E}_{\mathrm{min}}}^{{E}_{\mathrm{max}}}\displaystyle \frac{d\sigma }{{{dE}}_{\gamma }}\displaystyle \frac{{{dF}}_{{\rm{s}}}}{{{dE}}_{{\rm{s}}}}{{dE}}_{{\rm{s}}}\;=\;{{eE}}_{\parallel },\end{eqnarray} \tag{ 31 }$$

where ${m}_{{\rm{e}}}$ denotes the rest mass of the electron, E_γ and ${E}_{{\rm{s}}}$ the (emitted) hard-photon and (input) soft-photon energies, respectively, ${E}_{\mathrm{min}}$ and ${E}_{\mathrm{max}}$ the lower and upper bound energies we are considering, $d\sigma /{{dE}}_{\gamma }$ the Klein–Nishina differential cross section, and ${{dF}}_{{\rm{s}}}/{{dE}}_{{\rm{s}}}$ the differential flux of the background soft-photon field. Note that $d\sigma /{{dE}}_{\gamma }$ has ${\gamma }_{\mathrm{IC}}$ dependence. In Section 5, we demonstrate that the IC scatterings mainly take place in the extreme Klein–Nishina regime.

We assume that the input ADAF photon field (Mahadevan 1997) is isotropic in the BH rest frame. To cover the photon energies from radio wavelengths (of ADAF emission) to the VHE, we adopt ${E}_{\mathrm{min}}\;=\;5.11\times {10}^{-6}$ eV and ${E}_{\mathrm{max}}\;=\;5.11\times {10}^{17}$ eV, dividing the energies logarithmically into 198 bins. To estimate the normalization of the ADAF photon differential flux at the gap, we divide the total ADAF differential luminosity (Mahadevan 1997) by $4\pi {(10{r}_{{\rm{g}}})}^{2}$, assuming that the ADAF emission flux is homogeneous at $r\lt 10{r}_{{\rm{g}}}$ but reduces by the ${(r/10{r}_{{\rm{g}}})}^{-2}$ factor at $r\gt 10{r}_{{\rm{g}}};$ this assumption is reasonable because the sub-millimeter-IR photons, which most efficiently contribute for both IC scatterings and photon–photon pair creation, are emitted from the innermost region of ADAF, $r\approx 10{r}_{{\rm{g}}}$. We compute both photon–photon pair creation and IC scatterings, from the gap to a large enough Boyer–Lindquist radius, $r\;=\;60{r}_{{\rm{g}}}$. If we evaluated the ADAF flux at $r\;=\;6{r}_{{\rm{g}}}$ (instead of $10{r}_{{\rm{g}}}$), similar solutions would be obtained at smaller $\dot{m}$. Nevertheless, emission properties of the gap (e.g., spectrum near the critical accretion rate) are little changed by the normalization of the ADAF photon flux. Note that the charge-starvation condition ${n}_{\pm }/{n}_{\mathrm{GJ}}\lt 1$ (Equation (29)) is more easily satisfied if the ADAF flux is normalized at $r\;=\;10{r}_{{\rm{g}}}$ (than at $r\;=\;6{r}_{{\rm{g}}}$). See also Section 6.1.2 for the impact of the normalization of the soft photon field on the charge-starvation condition.

The actual Lorentz factor is computed by

$$\begin{eqnarray}&&{\gamma }^{-1}\;=\;{\gamma }_{\mathrm{curv}}{}^{-1}+{\gamma }_{\mathrm{IC}}{}^{-1},\end{eqnarray} \tag{ 32 }$$

which is almost equivalent with $\gamma \;=\;\mathrm{min}({\gamma }_{\mathrm{curv}},{\gamma }_{\mathrm{IC}})$.

At a Lorentz factor γ, a single electron emits (Rybicki & Lightman 1979)

$$\begin{eqnarray}&&{({P}_{\nu })}_{\mathrm{cv}}\;=\;\displaystyle \frac{\sqrt{3}{e}^{2}}{h\nu }\displaystyle \frac{\gamma }{{\rho }_{{\rm{c}}}}F\left(\displaystyle \frac{\nu }{{\nu }_{{\rm{c}}}}\right)\end{eqnarray} \tag{ 33 }$$

photons per unit frequency ν per unit time, where

$$\begin{eqnarray}&&F(x)\equiv x{\displaystyle \int }_{x}^{\infty }{K}_{5/3}(\xi )d\xi ,\end{eqnarray} \tag{ 34 }$$

$x\equiv \nu /{\nu }_{{\rm{c}}}$, and ${K}_{5/3}$ denotes the modified Bessel function of 5/3 order. The characteristic frequency is defined as

$$\begin{eqnarray}&&{\nu }_{{\rm{c}}}\equiv \displaystyle \frac{3}{4\pi }\displaystyle \frac{{\gamma }^{3}c}{{\rho }_{{\rm{c}}}}.\end{eqnarray} \tag{ 35 }$$

Particles have different Lorentz factors at different positions. We compute the evolution of γ as a function of position, assuming that electrons are created homogeneously at each position. If pairs are created at $s\;=\;{s}_{\mathrm{cr}}$, the electrons are accelerated outward to attain the Lorentz factor at s,

$$\begin{eqnarray}&&\gamma (s)\;=\;\mathrm{min}\left(\displaystyle \frac{-e}{{m}_{{\rm{e}}}{c}^{2}}{\displaystyle \int }_{{s}_{\mathrm{cr}}}^{s}{E}_{\parallel }(s){ds},{\gamma }_{\mathrm{curv}}\right).\end{eqnarray} \tag{ 36 }$$

To count up the photons emitted from various positions, we divide the full gap width into n_x parts along the particle path. Accordingly, we can compute the number of photons emitted outwardly in the ith energy bin per unit time by

$$\begin{eqnarray}&&{({N}_{\mathrm{cv}})}_{i}\;=\;\displaystyle \frac{{N}_{{\rm{e}}}}{{n}_{x}}{\displaystyle \int }_{{\nu }_{i-1}}^{{\nu }_{i}}{({P}_{\nu })}_{\mathrm{cv}}d\nu ,\end{eqnarray} \tag{ 37 }$$

where ${N}_{{\rm{e}}}$ denotes the total number of electrons existing in the entire gap, and ${\nu }_{i-1}$ and ${\nu }_{i}$ the lower and upper bound of the ith photon energy bin. In Section 4.4, we will describe how ${N}_{{\rm{e}}}$ is computed.

Next, we consider the emission properties of the IC process. At a Lorentz factor γ, a single electron emits

$$\begin{eqnarray}&&{({P}_{\nu })}_{\mathrm{IC}}\;=\;\displaystyle \frac{3{\sigma }_{{\rm{T}}}}{16}(1-\beta {\mu }_{{\rm{c}}}){\displaystyle \int }_{0}^{\infty }{{dE}}_{{\rm{s}}}\displaystyle \frac{{{dF}}_{{\rm{s}}}}{{{dE}}_{{\rm{s}}}}{\displaystyle \int }_{-1}^{1}{{dx}}^{*}{\displaystyle \int }_{0}^{2\pi }d{\alpha }^{*}{f}_{\mathrm{IC}}\end{eqnarray} \tag{ 38 }$$

photons per unit frequency per unit time, where ${f}_{\mathrm{IC}}\;=\;{f}_{\mathrm{IC}}({x}^{*},{\alpha }^{*},{E}_{{\rm{s}}},{\mu }_{{\rm{c}}},\gamma )$ is defined by Equations (26)–(28) of Hirotani et al. (2003), and ${\mu }_{{\rm{c}}}$ denotes the cosine of the collision angle. Assuming isotropic IC scatterings in the BH rest frame when we average over the collision angles, we obtain the number of photons emitted outwardly in the ith energy bin per unit time as follows:

$$\begin{eqnarray}&&{({N}_{\mathrm{IC}})}_{i}\;=\;\displaystyle \frac{{N}_{{\rm{e}}}}{{n}_{x}}{\displaystyle \int }_{{\nu }_{i-1}}^{{\nu }_{i}}d\nu \displaystyle \frac{1}{2}{\displaystyle \int }_{-1}^{1}{({P}_{\nu })}_{\mathrm{IC}}d{\mu }_{{\rm{c}}}.\end{eqnarray} \tag{ 39 }$$

The γ-rays emitted via curvature or IC process are partly absorbed by colliding with the ADAF soft photons. The absorption optical depth for the photons in the ith energy bin becomes

$$\begin{eqnarray}&&{\tau }_{i}\;=\;\displaystyle \frac{{s}_{2}-{s}_{1}}{c}\cdot \displaystyle \frac{1}{2}{\displaystyle \int }_{-1}^{1}d{\mu }_{{\rm{c}}}{\displaystyle \int }_{{E}_{\mathrm{th}}}^{\infty }{\sigma }_{{\rm{p}}}({E}_{i},{E}_{{\rm{s}}},{\mu }_{{\rm{c}}})\displaystyle \frac{{{dF}}_{{\rm{s}}}}{{{dE}}_{{\rm{s}}}}{{dE}}_{{\rm{s}}},\end{eqnarray} \tag{ 40 }$$

where ${\sigma }_{{\rm{p}}}$ refers to the photon–photon pair-creation mean-free path (Jauch & Rohrlich 1955), and ${E}_{i}\;=\;h({\nu }_{i-1}+{\nu }_{i})/2$ denotes the γ-ray energy. The threshold energy is given by ${E}_{\mathrm{th}}\;=\;[2/(1-{\mu }_{{\rm{c}}})]{({m}_{{\rm{e}}}{c}^{2})}^{2}/{E}_{i}$. The photon–photon collision is assumed to be isotropic. Since the typical energy of the curvature photons is around TeV, only the ADAF photons above near-IR frequency contribute for pair creation. However, at $\dot{m}\lt {10}^{-4}$, the number flux of these photons is typically eight orders of magnitude smaller than the sub-millimeter-IR photons, which will effectively collide with ∼100 TeV IC photons. Thus, for low-luminosity AGNs, curvature photons do not effectively materialize as pairs in the BH gap, forming a striking contrast to pulsar gaps. This point will be discussed in Section 5 in detail.

Although the curvature photons do not materialize in a BH gap of a low-luminosity AGN, we consider both the curvature and IC processes for completeness. The number of pairs cascaded from a single electron is given by

$$\begin{eqnarray}&&{ \mathcal M }\;=\;{{\rm{\Sigma }}}_{i}[{({N}_{\mathrm{cv}})}_{i}+{({N}_{\mathrm{IC}})}_{i}][1-\mathrm{exp}(-{\tau }_{i})],\end{eqnarray} \tag{ 41 }$$

where i denotes the photon energy bin. For simplicity, we assume that the same ${ \mathcal M }$ can be applied for both the outgoing electrons and the ingoing positrons. For a gap to be sustained stationarily, multiplicity ${ \mathcal M }$ should be unity. That is, a single electron materializes into ${ \mathcal M }$ pairs on average within the gap. The returned ${ \mathcal M }$ positrons emit copious γ-rays inwards, ${{ \mathcal M }}^{2}$ of which materialize as pairs. As a result, a single electron cascades into ${{ \mathcal M }}^{2}$ electrons after one "round trip." We thus obtain ${ \mathcal M }\;=\;1$ as the condition for a gap to be sustained stationarily. If this inward–outward symmetry breaks down, we must impose a more complicated gap closure condition that the product of the inward and outward multiplicities should become unity (Hirotani 2013).

From the condition of ${ \mathcal M }\;=\;1$, we can update s₂. We solve Equation (9) from $s\;=\;{s}_{2}$ to the inner boundary $s\;=\;{s}_{1}$ where ${E}_{\parallel }$ vanishes. The inner and outer boundaries, $s\;=\;{s}_{1}$ and $s\;=\;{s}_{2}$, are determined as a free-boundary problem because we impose the gap closure condition, ${ \mathcal M }=\infty$ and ${{dE}}_{\parallel }/{ds}$ at $s\;=\;{s}_{2}$ by specifying j through Equations (9) and (10). The updated ${E}_{\parallel }(s)$ is then substituted into Equations (30) and (31) to update $\gamma (s)$. Subsequently, ${({N}_{\mathrm{CV}})}_{i}$ and ${({N}_{\mathrm{IC}})}_{i}$ are updated by (33), (35), and (37)–(39). Then ${ \mathcal M }\;=\;1$ (Equation (41)) updates s₂ again. We iterate this process until the quantities converge.

Let us first evaluate the magnetic field strength with Equation (24). Substituting $\dot{m}\;=\;{10}^{-4}$ and M₉ = 0.3 into Equation (24), we find the equilibrium magnetic field strength, $B\;=\;{B}_{\mathrm{eq}}\;=\;7.3\times {10}^{2}$ G. Thus, Equation (23) gives ${L}_{\mathrm{sd}}\approx 5.3\times {10}^{41}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$. We should notice here that the Blandford–Znajek power, ${L}_{\mathrm{sd}}$, gives the upper limit of the gap luminosity because the BH gap is only liberating a part of the rotational energy of the BH. Near the BH, the magnetic field tends to be radial (or split-monopole-like) due to the inertia of the plasmas and the causality at the horizon (Hirotani et al. 1992; McKinney et al. 2012). If we sum up the outward emission along such (quasi) radial field lines over the entire null surface, the resultant photon flux will become more or less isotropic. In this case, the photon flux to be detected little depends on the beaming angles of the emitted photons.

In contrast to ${L}_{\mathrm{gap}}\lt 5.3\times {10}^{41}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ mentioned just above, MAGIC detected the isotropic luminosity of $\sim 2\times {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ during the flare (Aleksić et al. 2014b). Thus, we must conclude that the BH gap can reproduce at most 0.3 % of the observed flare luminosity of IC 310 if $B\leqslant {B}_{\mathrm{eq}}$. Since the gap will be quenched by the ADAF-provided pairs at $\dot{m}\gt {10}^{-4}$, there is no hope to obtain such a large gap luminosity as ${L}_{\mathrm{gap}}\approx 2\times {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ (Equation (21)), if we adopted the equilibrium magnetic field for IC 310.

To explore the possibility of a stronger B, let us consider a magnetized plasma around an extremely rotating BH, ${r}_{{\rm{g}}}\gt a\geqslant 0.998{r}_{{\rm{g}}}$ (Bardeed 1970; Thorne 1974). If a BH is extremely rotating, causality does not allow an additional accretion of plasmas that have positive angular momenta, thereby halting accretion at the event horizon. As a result of this cancellation of the gravity by the centrifugal force, a strong magnetic field can be confined by the accumulated, dense plasmas. For instance, magnetohydrodynamic accretions have a vanishing Alfvénic Mach number, ${M}_{{\rm{A}}}\propto {{Bn}}^{-1/2}$ (Camenzind 1986a, 1986b), and hence a diverging plasma density n, at the event horizon in the limit of $a\to {r}_{{\rm{g}}}$ (Hirotani et al. 1992). In addition, the GR MHD simulation shows that the magnetic energy becomes approximately 30 times (or 100 times) greater for $a\;=\;0.998{r}_{{\rm{g}}}$ near the horizon, compared to the $a\;=\;0.9{r}_{{\rm{g}}}$ (or a = 0) case under similar initial and boundary conditions (Hirose et al. 2004). This enhancement of the magnetic field is due to a plasma compilation near the horizon. The enhanced magnetic field enters well within the ergosphere, having a non-negligible radial component in the funnel. Thus, in the higher latitudes, it is capable of extracting the hole's rotational energy efficiently.

To set up the model as simple as possible, we therefore assume $B\approx {10}^{4}$ G irrespective of $\dot{m}$ under ${r}_{{\rm{g}}}\gt a\geqslant 0.998{r}_{{\rm{g}}}$ in this paper. Although Equation (23) gives ${L}_{\mathrm{sd}}\approx {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ in this case, which greatly exceeds the observed jet power by 50 times, we expect that such a strong magnetic field, $B\gt 14{B}_{\mathrm{eq}}$, can be sustained near the horizon only in a much shorter period than the jet propagation timescale. In other words, the BH gap of IC 310 is activated only intermittently with a small duty cycle, e.g., $\sim 1/50\;=\;0.02$. Although it is important to confirm the feasibility of such a strong B around an extremely rotating BH by a numerical analysis, such an investigation is irrelevant to the main subject of the present paper. If B greatly exceeded 10⁴ G, on the other hand, the BHs energy and angular momentum would be electromagnetically extracted so efficiently that the extracted power would exceed the jet power of typical AGNs (Ghisellini et al. 2014). On these grounds, we assume that the BH is extremely rotating, $a\;=\;0.998{r}_{{\rm{g}}}$, and that $B\approx {10}^{4}$ G holds during the flare. For such an extreme Kerr BH, the horizon radius ${r}_{{\rm{H}}}$ becomes about a half of the Schwarzshild radius.

If the BH is extremely rotating at $a\approx 0.998{r}_{{\rm{g}}}$, the null surface is located at radius $r\;=\;{r}_{0}\approx 2.1{r}_{{\rm{H}}}$ in $0\leqslant \theta \lt \pi /2$ (Figure 1) if ${{\rm{\Omega }}}_{{\rm{F}}}\approx 0.3{\omega }_{{\rm{H}}}$, irrespective of the field-line geometry. In what follows, unless explicitly mentioned, we adopt $\theta \;=\;15^\circ$ (Aleksić et al. 2014b), and assume a split-monopole magnetic field (left panel of Figure 1). Note that the argument is subject to change only mildly if we adopt different field-line geometry such as a parabolic one (right panel of Figure 1), particularly at higher latitudes, $\theta \lt 30^\circ$.

The full gap width $w\;=\;{s}_{2}-{s}_{1}$ increases with decreasing $\dot{m}$ as depicted in the top panel of Figure 8. If the accretion rate decreases to $4.87\times {10}^{-6}$, ${r}_{0}+{s}_{1}\to {r}_{{\rm{H}}}\;=\;1.06{r}_{{\rm{g}}}$, that is, the inner boundary eventually touches down the horizon. Below this critical accretion rate, no gap solution exists.

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In the middle panel, we present ${\gamma }_{\mathrm{curv}}$ (Equation (30)), ${\gamma }_{\mathrm{IC}}$ (Equation (31)), and γ (Equation (32)) with dashed, thick-dotted, and solid curves. Note that the IC drag force is explicitly computed by fully taking account of the Klein–Nishina effect. For reference, we also plot the IC-limited Lorentz factor ${\gamma }_{\mathrm{Thomson}}$ that would be obtained if the scatterings take place in the Thomson regime with thin dotted curve. It is clear that the IC process takes place in the extreme Klein–Nishina regime and that the deviation from ${\gamma }_{\mathrm{Thomson}}$ becomes prominent at $\dot{m}\ll {10}^{-4}$. Since we compute ${\gamma }_{\mathrm{Thomson}}$ using a single parameter in the same way as LR11 for reference purpose (instead of replacing the Klein–Nishina cross section with the Thomson cross section in Equation (31)), the thick and thin dotted curves do not match around $\dot{m}\sim {10}^{-4}$. It also shows that the Lorentz factors are limited by the curvature process (Equation (30)) in almost the entire range of $\dot{m}\lt {10}^{-4}$. Since the curvature drag force dominates the IC one, the gap luminosity is dominated by the curvature process. Nevertheless, explicit computation of the IC emission with the Klein–Nishina cross section is, indeed, essential. This is because the closure condition, ${ \mathcal M }\;=\;1$, is satisfied by the IC photons, or equivalently because the curvature photons do not materialize as pairs colliding with ADAF soft photons. In short, the Lorentz factors are limited by ${\gamma }_{\mathrm{curv}}$ in the entire range of $\dot{m}\lt {10}^{-4}$.

In the bottom panel, we plot the number density of the pairs created outside the gap via photon–photon collisions. Since a force-free magnetosphere can be sustained when the created pair density exceeds the GJ value, the present solutions give natural stationary solutions of the gap. The created pair density exceeds the GJ value by more than two orders of magnitude when the gap becomes most luminous because $\sim {10}^{4}$ TeV IC photons cascade into $\;{10}^{2}$ TeV pairs. Since the charge density becomes comparable to the GJ value in the gap when $w\gg {r}_{{\rm{g}}}$, it means that the multiplicity for a primary lepton to cascade into secondaries (and higher generation pairs) is also $\sim {10}^{2}$. Since we are assuming $B\;=\;{10}^{4}$ G irrespective of $\dot{m}$, the GJ density (horizontal dotted line) is constant with $\dot{m}$ in the present treatment.

Let us examine the spatial distribution of the gap. Since the ADAF MeV photon density will be more or less homogeneous, we apply $\dot{m}\;=\;3.16\times {10}^{-5}$ for all the field lines. For illustration purposes, we adopt ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;0.3{\omega }_{{\rm{H}}}$ for all the field lines as a practical compromise. It is, however, noteworthy that $-0.1{{\rm{\Omega }}}_{{\rm{H}}}\lt {{\rm{\Omega }}}_{{\rm{F}}}\lt 0.35{{\rm{\Omega }}}_{{\rm{H}}}$ is indicated in the funnel by the numerical simulations and the supporting analytical work, both of which are claimed to represent radio-loud AGNs (McKinney et al. 2012; Beskin & Zheltoukhov 2013).

At $\dot{m}\;=\;3.16\times {10}^{-5}$, the gap distributes on the meridional plane as depicted by the green region in Figure 9. At $\theta \;=\;15^\circ$, 45°, and 75°, the full gap thickness (along the magnetic field line on the poloidal plane) becomes $w\;=\;{s}_{2}-{s}_{1}\;=\;0.525{r}_{{\rm{g}}}$, $0.538{r}_{{\rm{g}}}$, and $0.593{r}_{{\rm{g}}}$, respectively (see also the top panel of Figure 8). Since ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;{{\rm{\Omega }}}_{{\rm{F}}}({\rm{\Psi }})$ is assumed to be constant here, the null surface distributes nearly spherically. In this case, the gap solution little changes in the higher latitudes because ${\rho }_{\mathrm{GJ}}$ distribution does not have a strong dependence on θ in $\theta \lt 30^\circ$. Therefore, we assume that the split-monopole magnetic field extends up to $60{r}_{{\rm{g}}}$ to compute the pair cascade outside the gap. This treatment is probably justified in the polar funnel of accreting BH systems (McKinney & Narayan 2007a, 2007b).

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Computing the real charge distribution with Equation (10), we obtain ${E}_{\parallel }$ for the critical accretion rate, $\dot{m}\;=\;4.87\times {10}^{-6}$, below which no gap solution exists. The result is depicted in Figure 10, where the event horizon is located at $s\;=\;-1.09{r}_{{\rm{g}}}$ on the abscissa. It follows that ${E}_{\parallel }$ maximizes around the middle point between the horizon and the null surface (s = 0). Therefore, a significant fraction of the VHE photons are emitted within $2{r}_{{\rm{g}}}\lt {r}_{0}\approx 2.1{r}_{{\rm{g}}}$ via the curvature process. The maximum attainable Lorentz factor in the middle panel of Figure 8 is computed from the solved ${E}_{\parallel }(s)$ at each $\dot{m}$, as depicted in the lower panel of Figure 10.

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In Figure 11, we depict the spectra of the photons emitted by the outgoing electrons as thick lines, for three discrete accretion rates, $\dot{m}$. The outgoing particles consist of the primary electrons accelerated in the gap and the cascaded pairs outside the gap. These particles upscatter the background soft photons, which will be partly absorbed by the same soft-photon field. The soft photons are provided by the ADAF, whose corresponding spectra (Mahadevan 1997) are indicated by the thin curves. In the $\dot{m}\to 0$ limit, ${{dF}}_{{\rm{s}}}/{{dE}}_{{\rm{s}}}\to 0$ means $w\;=\;{s}_{2}-{s}_{1}\to \infty$ in Equation (40) in order that ${ \mathcal M }\;=\;1$ may be held in Equation (41). Thus, ${r}_{0}+{s}_{1}$ should approach the lower limit, ${r}_{{\rm{H}}}$, if $\dot{m}$ reduces enough. In the present case, ${r}_{0}+{s}_{1}$ reaches ${r}_{{\rm{H}}}$, when $\dot{m}\;=\;4.87\times {10}^{-6}$ (upper panel of Figure 8). This most luminous case is depicted as the red thick line in Figure 11. We find that the flux of the flare (red filled circles in the inset) can be qualitatively reproduced. It looks contradictory that the gap becomes more luminous with decreasing accretion rate. However, it is a natural consequence of rotation-powered particle accelerator such as in pulsars. In the present case, it is the BH's rotational energy that energizes the gap. It forms a striking contrast to accretion-powered systems.

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Figure 11 also shows that the VHE spectrum extends into higher energies with increasing VHE flux. This is because the soft-photon density decreases with decreasing $\dot{m}$, thereby reducing the absorption of the primary IC photons above 10 TeV. Owing to these photon–photon collisions, there appears a sharp cutoff around 50–100 TeV. In the most luminous case (red solid line), the cutoff appears around 100 TeV.

Fermi-LAT observed IC 310 between 2008 August 5, and 2011 July 31, and detected eight photons between 12 and 148 GeV (Aleksić et al. 2014a); the fluxes are plotted by the open circles. Considering the small duty cycle of the IC 310's BH gap, the Fermi-LAT flux, which was averaged over three years, appears to be higher than our prediction. In addition, the X-ray fluxes, which are represented by the solid and dotted bowties, obviously exceed our prediction. The strong X-ray flux may be due to the emission from a corona or the jet.

In Figure 12, we plot the total, primary IC, primary curvature spectra as the thick red solid, black dashed, and black dash-dotted lines for the case of critical accretion rate, $\dot{m}\;=\;4.87\times {10}^{-6};$ the red lines are common in Figures 11 and 12. Since the mean-free path of IC scatterings is much longer than w, the highest energy photons ($\gt 100$ TeV) are mostly emitted outside the gap. On the contrary, since the electrons cool down at much shorter intervals than w via the curvature process, the curvature photons are mostly emitted inside the gap. These curvature photons dominate the 0.1–10 TeV flux, becoming more and more important with decreasing $\dot{m}$, and hence with increasing output flux. This is because the curvature emission power is proportional to ${\gamma }^{4}$, while the IC one is proportional to ${\gamma }^{1}\sim {\gamma }^{2}$ (depending on whether the scatterings take place in the extreme Klein–Nishina or the Thomson limit). Since the absorption works mainly above 10 TeV, the curvature photons are little absorbed by the ADAF soft-photon field.

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The primary IC photons are, on the contrary, heavily absorbed to be reprocessed as the secondary synchrotron and IC components, which are depicted as the dash-dot-dot-dotted line. These secondary photons are absorbed again above 10 TeV to be reprocessed as the tertiary component (black dotted line). We compute such a pair-creation cascade until the 30th generation, depicting the quaternary and quinary components as the green dotted lines (as labeled with 4 and 5).

Let us briefly examine the case in which B is evaluated by its equilibrium value, Equation (24). In Figure 13, we present the photons spectra. The input parameters are the same as Figure 11. However, the critical accretion rate becomes $\dot{m}\;=\;5.62\times {10}^{-6}$ in this case. As analytically expected in Section 5.1, the weaker magnetic field, B = 43.3 G at this critical accretion rate cannot reproduce the observed VHE fluxes at all.

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It may be desirable to sum up the main points that have been made in this Section 5.6.

• The primary electrons (or positrons) are accelerated outward (or inward) by the magnetic-field-aligned electric field.
• The primary particles emit primary photons efficiently via the IC and the curvature processes.
• If these primary photons materialize as pairs in the gap, the created pairs are separated and accelerated to become the primary particles as listed in item (1a).
• The observed VHE flare photons are mostly emitted via the curvature process.
• Although the curvature process dominates the IC one, it is the IC photons that materialize within the gap, colliding with the ADAF submillimeter photons. Thus, both curvature and IC processes are essential, forming a striking contrast to pulsar gap models, in which only the curvature process governs the electrodynamics (because of the higher soft-photon energies).
• If these primary photons materialize outside the gap, the created pairs migrate away from the gap as the secondary pairs.
• These secondary pairs emit secondary photons via the IC and the synchrotron processes.
• If these secondary photons materialize within the magnetosphere, the created pairs migrate away from the gap as the tertiary pairs.
• These tertiary pairs emit tertiary photons via IC and synchrotron processes, and so on.
• We recur such calculations until the pairs or the photons propagate to $60{r}_{{\rm{g}}}$, or until the pairs cascade into the 30th generation.

Let us look in more detail at the pairs cascaded outside the gap. To examine the most efficient case of pair cascade, we consider the solution at the critical accretion rate, $\dot{m}\;=\;4.87\times {10}^{-6}$ for $B\;=\;{10}^{4}{({r}_{0}/r)}^{2}$ G (i.e., the red curves in Figures 11 and 12). Figure 14 shows the pairs' energy spectra, integrated over the entire volume of the BH magnetosphere within ${r}_{0}+{s}_{2}\lt r\lt 60{r}_{{\rm{g}}}$, assuming a spherical symmetry. The primary IC photons collide with the radio-submillimeter ADAF photons to materialize as the secondary pairs whose energy spectrum is represented by the red solid line. Such secondary pairs upscatter the ADAF soft photons that are capable of materializing into the tertiary pairs (black dashed). Such tertiary pairs emit tertiary γ-rays, some of which materialize as the quaternary pairs (green dash-dotted). At such a low accretion rate as $\dot{m}\;=\;4.87\times {10}^{-6}$, pair-creation cascade finishes at the quinary generation (blue dotted). The spectrum sharpens with increasing generation. The reasons are twofold:

• i.  
  The energy of the cascading pairs decreases with increasing generation.
• ii.  
  The higher generation pairs are created at greater r where the ADAF photon intensity reduces by a ${r}^{-2}$ law. Thus, only relatively higher energy pairs can cascade into higher generations because lower energy pairs can emit only the lower energy IC photons that cannot materialize at greater r.

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In this way, the energy range is restricted from both higher and lower sides, showing a peaking profile around 100 TeV above the quaternary generation.

Typically speaking, primary IC photons around 10⁴ TeV cascade into 10² TeV pairs, which means that a single primary electron eventually cascades into $\sim {10}^{2}$ pairs in the BH magnetosphere. See also LR11 for similar conclusions.

Integrating over the pair energies, we obtain $3.03\times {10}^{41}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ as the kinetic luminosity of the created pairs. Since the primary ICS luminosity is $6.97\times {10}^{41}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, 43% of the upscattered photon energy is converted into cascaded pairs outside the gap. However, it is only 0.12% of the primary curvature luminosity.

Let us investigate how the solution depends on the curvature radius, ${\rho }_{{\rm{c}}}$. In Figures 15 and 16, we plot the spectrum obtained for ${\rho }_{{\rm{c}}}\;=\;0.1r$ and $10r$ at radius r, respectively. The green and black curves represent the cases of $\dot{m}\;=\;1.00\times {10}^{-4}$ and $3.16\times {10}^{-5}$, in the same manner as in Figure 11. The red curve corresponds to the critical accretion rate; $\dot{m}\;=\;7.50\times {10}^{-6}$ and $\dot{m}\;=\;5.62\times {10}^{-6}$ for Figures 15 and 16, respectively. It follows that the curvature spectrum, which peaks between 0.1 and 1 TeV, becomes softer with decreasing ${\rho }_{{\rm{c}}}$. This is because Equations (30) and (35) give ${\nu }_{{\rm{c}}}\propto {\rho }_{{\rm{c}}}{}^{1/2}$, which shows that the curvature-emitted photon energy decreases an order of magnitude if ${\rho }_{{\rm{c}}}$ decreases from $10r$ to $0.1r$. The inset of Figure 15 shows that the flare spectrum is difficult to be reproduced above 1 TeV, if the curvature radius becomes as small as $\sim 0.1r$. In this case, electrons' Lorentz factors saturate at one order-of-magnitude smaller values than the dashed curve in the middle panel of Figure 8 because ${E}_{\parallel }$ in Equation (30) depends on ${\rho }_{{\rm{c}}}$ only weakly in the vicinity of a rotating BH, unlike the situation around a rotating neutron star (NS). Note that $\partial {\rho }_{\mathrm{GJ}}/\partial s$ is due to the frame-dragging effect in BH magnetospheres, whereas it is due to the global field-line curvature in NS magnetospheres. Thus, ${E}_{\parallel }$ little depends on ${\rho }_{{\rm{c}}}$ in BH magnetospheres.

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In a time-averaged sense, in the polar funnel, the magnetic field lines threading the horizon are well ordered and approximated by a split-monopole configuration (McKinney & Narayan 2007a; McKinney et al. 2012). Although the split-monopole field is tightly wound azimuthally due to frame dragging near the horizon, it is predominantly radial with loosely wound helics moderately away from the horizon (e.g., at $r\gt 2{r}_{{\rm{g}}}$; Hirose et al. 2004). Thus, we consider that ${\rho }_{{\rm{c}}}\sim 10r$ may be possible in a time-averaged sense.

Despite the constancy of time-averaged magnetized accretion, the instantaneous accretion rate varies considerably with time. In addition, the instantaneous magnetic field lines are kinked and bent in the evacuated funnel, which becomes a cauldron of strong waves (Krolik et al. 2005). Thus, it may be reasonable to consider a superposition of the spectra obtained under different ${\rho }_{{\rm{c}}}$'s. For instance, due to a fluctuating magnetic field structure, photons may be emitted into various directions from each position in a time-dependent manner. In such a case, we will detect the combined spectra from different field lines with various ${\rho }_{{\rm{c}}}$.

As an example, in Figure 17, we present the result for the superposition of the ${\rho }_{{\rm{c}}}\;=\;0.1r$ (blue dotted) and ${\rho }_{{\rm{c}}}\;=\;10r$ (red dashed) cases with weights of 0.25 and 0.75, respectively. Since the ADAF photon field may be inhomogeneous (Li et al. 2008), we adopt here different $\dot{m}$ for the ${\rho }_{{\rm{c}}}\;=\;0.1r$ and $10r$ cases; namely, $\dot{m}\;=\;7.50\times {10}^{-6}$ for the blue dotted line, and $\dot{m}\;=\;5.62\times {10}^{-6}$ for the red dashed line. That is, the blue dotted line coincides with the red solid line in Figure 15, whereas the red dashed line coincides with the red solid line in Figure 16. It follows that a power-law-like spectrum can be formed if curvature spectra with varying ${\rho }_{{\rm{c}}}$ are superposed with appropriate weights. So that the observed power-law-like VHE spectrum may be reproduced, greater weights are favorable at greater ${\rho }_{{\rm{c}}}$. In other words, the observed flare spectrum may imply that the instantaneous magnetic field lines are mostly straight but kinked moderately in the funnel. The actual distribution of the weight as a function of ${\rho }_{{\rm{c}}}$ should be examined separately by numerical analysis. However, it is out of the scope of the present paper.

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Let us finally examine the dependence on ${{\rm{\Omega }}}_{{\rm{F}}}$. First, we consider the distribution of the dimensionless GJ charge density, ${\rho }_{\mathrm{GJ}}/({{\rm{\Omega }}}_{{\rm{F}}}B/2\pi c)$, on the poloidal plane, assuming a split-monopole magnetic field-line configuration. In the left panel of Figure 18, we adopt a slower field-line rotation, ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;0.15{\omega }_{{\rm{H}}}$. It follows that the null surface (red solid curve) shifts to $r\approx 2.9{r}_{{\rm{g}}}$ from $\approx 2.1{r}_{{\rm{g}}}$ as ${{\rm{\Omega }}}_{{\rm{F}}}$ reduces to $0.15{\omega }_{{\rm{H}}}$ from $0.30{\omega }_{{\rm{H}}}$ (cf. the left panel of Figure 1). This is because ω reduces by ${r}^{-3}$, which indicates that $\omega \;=\;{{\rm{\Omega }}}_{{\rm{F}}}$ is realized at greater r for a smaller ${{\rm{\Omega }}}_{{\rm{F}}}$. We plot the (red) dashed contours only below ${\rho }_{\mathrm{GJ}}/({{\rm{\Omega }}}_{{\rm{F}}}B/2\pi c)\;=\;2.0$ because the gap inner boundary is located in the region where ${\rho }_{\mathrm{GJ}}/({{\rm{\Omega }}}_{{\rm{F}}}B/2\pi c)\lt 2.0$ holds, unless a super-GJ electric current is injected across the outer boundary. On the other hand, if the magnetic field lines rotate much faster as ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;0.60{\omega }_{{\rm{H}}}$, the null surface is located at $r\approx 1.5{r}_{{\rm{g}}}$, as the right panel shows. In this case, the inner boundary more easily touches down the horizon because the GJ charge density at the horizon, $| {\rho }_{\mathrm{GJ}}{| }_{{\rm{H}}}$ is proportional to ${\omega }_{{\rm{H}}}-{{\rm{\Omega }}}_{{\rm{F}}}$, thereby being limited at a smaller value if ${{\rm{\Omega }}}_{{\rm{F}}}$ approaches ${\omega }_{{\rm{H}}}$.

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Second, we consider the γ-ray spectrum. At ${{\rm{\Omega }}}_{{\rm{F}}}\lt 0.30{\omega }_{{\rm{H}}}$, we can obtain a similar VHE spectrum as the $0.30{\omega }_{{\rm{H}}}$ case when the inner boundary does not touch down the horizon, provided that B is unchanged at 10⁴ G in the gap. Alternatively, we obtain a similar VHE spectrum for $B\lt {10}^{4}$ G when the inner boundary touches down the horizon. On the other hand, at ${{\rm{\Omega }}}_{{\rm{F}}}\gt 0.30{\omega }_{{\rm{H}}}$, we cannot obtain the observed flare VHE flux even when the inner boundary touches down the horizon, provided that $B\;=\;{10}^{4}$ G. We could adopt $B\gt {10}^{4}$ G and obtain a similar VHE spectrum; however, a greater B would result in a further greater jet-power efficiency (see Section 6.1), which is not favored. In Figure 19, we present the spectra obtained for ${{\rm{\Omega }}}_{{\rm{F}}}\;=\;0.60{\omega }_{{\rm{H}}}$ and $B\;=\;{10}^{4}$ G. It shows that the VHE flux reduces significantly and that the curvature spectrum softens if ${{\rm{\Omega }}}_{{\rm{F}}}\gt 0.3{\omega }_{{\rm{H}}}$ (cf. Figure 11). Thus, we can conclude that a smaller value of ${{\rm{\Omega }}}_{{\rm{F}}}$ in the polar funnel, which is suggested from numerical and analytical works (McKinney et al. 2012; Beskin & Zheltoukhov 2013), is favorable to obtain a greater VHE flux from a BH gap.

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6.1.1. Energy Extraction Efficiency

6.1.2. Photon Density Constrained by X-Ray Observations

In this paper, we have examined the ADAF theory and concluded that the BH magnetosphere of IC 310 becomes charge starved if $\dot{m}\lt {10}^{-4}$. To check this logic, in this subsection, we compare the MeV photon density predicted by ADAF (Equation (26)),

$$\begin{eqnarray}&&{n}_{{\rm{s}}}\;=\;5.64\times {10}^{4}{\left(\displaystyle \frac{{M}_{9}}{0.3}\right)}^{-1}{\left(\displaystyle \frac{\dot{m}}{2\times {10}^{-4}}\right)}^{2}{\left(\displaystyle \frac{r}{6{r}_{{\rm{g}}}}\right)}^{-2}{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 42 }$$

with observations.

Unfortunately, there are no direct observations around MeV for IC 310 along with other low accretion rate BH systems. Thus, to estimate the MeV photon density, here we utilize the X-ray observations and extrapolate the spectrum into MeV energies.

The X-ray spectrum of IC 310 has been obtained between 2 and 10 keV with XMM-Newton (MJD 52697, i.e., 2003 February 27), Chandra (MJD 53363, 53456), and Swift/XRT (MJD 54152) (Sato et al. 2005; Aleksić et al. 2014a). The highest X-ray flux $2.5\times {10}^{-3}\;\mathrm{keV}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}$ (with photon index ${\rm{\Gamma }}\;=\;2.01$) was obtained by Chandra (MJD 55456), while the lowest one $8.28\times {10}^{-4}\;\mathrm{keV}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}$ (with photon index ${\rm{\Gamma }}\;=\;2.55$) was obtained by XMM-Newton. We will consider both cases in this subsection to estimate the lower and upper bound of the photon number density around MeV.

First, to extrapolate the observed power-law spectrum into MeV, we assume an exponential cutoff above energy ${E}_{\mathrm{br}}$, in the differential photon number flux,

$$\begin{eqnarray}&&\displaystyle \frac{{df}}{{dE}}\;=\;{{NE}}^{-{\rm{\Gamma }}}\mathrm{exp}\left(-\displaystyle \frac{E}{{E}_{\mathrm{br}}}\right)\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\;{\mathrm{keV}}^{-1},\end{eqnarray} \tag{ 43 }$$

where the photon energy E is measured in keV. Multiplying E and integrating Equation (43) from 2 to 10 keV, we obtain the energy flux in 2–10 keV,

$$\begin{eqnarray}&&{F}_{{\rm{X}}}\;=\;N\displaystyle \frac{{10}^{2-{\rm{\Gamma }}}-{2}^{2-{\rm{\Gamma }}}}{2-{\rm{\Gamma }}}\;\mathrm{keV}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}.\end{eqnarray} \tag{ 44 }$$

Setting ${F}_{{\rm{X}}}\;=\;2.5\times {10}^{-3}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\;{\mathrm{keV}}^{-1}$ and $8.28\times {10}^{-4}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\;{\mathrm{keV}}^{-1}$, we obtain

$$\begin{eqnarray}&&N\;=\;1.55\times {10}^{-3},\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&N\;=\;1.13\times {10}^{-3},\end{eqnarray} \tag{ 46 }$$

for the Chandra (MJD 55456) and XMM observations, respectively. We estimate the photon number density at MeV by

$$\begin{eqnarray}&&{n}_{{\rm{s}},\mathrm{obs}}\equiv {\left(\displaystyle \frac{d}{r}\right)}^{2}\displaystyle \frac{1}{c}{\left(E\displaystyle \frac{{df}}{{dE}}\right)}_{\mathrm{MeV}},\end{eqnarray} \tag{ 47 }$$

where d = 81.4 Mpc denotes the distance to IC 310, and

$$\begin{eqnarray}&&{\left(E\displaystyle \frac{{df}}{{dE}}\right)}_{\mathrm{MeV}}\;=\;{10}^{-3({\rm{\Gamma }}-1)}N\mathrm{exp}\left(-\displaystyle \frac{E}{{E}_{\mathrm{br}}}\right)\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\end{eqnarray} \tag{ 48 }$$

denotes the photon number flux at MeV. Thus, the photon number density at $r\;=\;6{r}_{{\rm{g}}}$ can be estimated to be

$$\begin{eqnarray}&&{n}_{{\rm{s}},\mathrm{obs}}\;=\;2.97\times {10}^{16-3{\rm{\Gamma }}}{{Ne}}^{-E/{E}_{\mathrm{br}}}{(r/6{r}_{{\rm{g}}})}^{-2}\;{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 49 }$$

For the low X-ray flux case of XMM observation, we obtain

$$\begin{eqnarray}&&{n}_{{\rm{s}},\mathrm{obs}}\;=\;8.8\times {10}^{5}{e}^{-E/{E}_{\mathrm{br}}}{(r/6{r}_{{\rm{g}}})}^{-2}\;{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 50 }$$

Therefore, we find that the e-folding energy ${E}_{\mathrm{br}}$ should be less than 370 keV so that ${n}_{{\rm{s}},\mathrm{obs}}$ may become less than ${n}_{{\rm{s}}}$ at the time-averaged accretion rate, $\dot{m}\;=\;2\times {10}^{-4}$. On the other hand, for the high X-ray flux case of the Chandra (MJD 55456) observation, we obtain

$$\begin{eqnarray}&&{n}_{{\rm{s}},\mathrm{obs}}\;=\;4.6\times {10}^{7}{e}^{-E/{E}_{\mathrm{br}}}{(r/6{r}_{{\rm{g}}})}^{-2}\;{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 51 }$$

Therefore, we find ${E}_{\mathrm{br}}\lt 140\;\mathrm{keV}$.

The temperature of Comptonizing electrons is estimated to be ${T}_{{\rm{e}}}\;=\;2{E}_{\mathrm{br}}$ for an optically thin plasma and ${T}_{{\rm{e}}}\;=\;3{E}_{\mathrm{br}}$ for an optically thick one (Petrucci et al. 2010). Thus, we obtain the conservative upper bound, ${T}_{{\rm{e}}}\lt 1.2\;\mathrm{MeV}$ and ${T}_{{\rm{e}}}\lt 420$ keV for the low and high X-ray flux cases, respectively. Considering that the Compton optical thickness is probably thin at small $\dot{m}$, ${T}_{{\rm{e}}}\lt 740$ and ${T}_{{\rm{e}}}\lt 280$ keV may be more appropriate as the upper bound.

Adopting an ADAF theory (Mahadevan 1997), one finds ${T}_{{\rm{e}}}\lt 600\;\mathrm{keV}$ at $\dot{m}\gt {10}^{-4}$. Therefore, the density of the MeV photon emitted by ADAF during the jet phase (not during the VHE flare phase) appear to be consistent with the typical MeV photon density deduced from the X-ray observations during the low X-ray flux phase. However, if we adopt the high X-ray flux phase, the required electron temperature, ${T}_{{\rm{e}}}\lt 280\;\mathrm{keV}$, appears to be too low. In addition, the e-folding energy is empirically deduced to be above 150 keV (i.e., ${T}_{{\rm{e}}}\gt 300\;\mathrm{keV}$ is obtained) commonly in low accretion rate systems (Del Santo 2012 for galactic BHs; Nowak et al. 2011 for Cyg X-1; Malizia et al. 2014 for Seyfert 1 s). Since ${T}_{{\rm{e}}}$, and hence ${E}_{\mathrm{br}}$ increases with decreasing $\dot{m}$, the MeV photon density may increase with decreasing $\dot{m}$ when $\dot{m}\ll {10}^{-4}$, although the normalization of the photon number density decreases in X-rays (e.g., in 2–10 keV). On these grounds, our ansatz of dropping the accretion rate by half in order to create a BH gap will be invalidated, if X-ray observations indicate a MeV photon density that exceeds Equation (42) for IC 310.

What is more, we used $r\;=\;6{r}_{{\rm{g}}}$ to evaluate ${n}_{{\rm{s}}}$ and ${n}_{{\rm{s}},\mathrm{obs}}$ in this subsection because the lower-limit radius of the ADAF emission is set to be ${r}_{\mathrm{min}}\;=\;6{r}_{{\rm{g}}}$ in the self-similar analytical solution we are adopting (Mahadevan 1997; see also e.g., Narayan et al. 1995; Lasota et al. 1996 for an accordant choice of ${r}_{\mathrm{min}}$). Note that the radius $r\;=\;6{r}_{{\rm{g}}}$ gives a conservative estimate (i.e., in this case, higher value) of ${n}_{{\rm{s}},\mathrm{obs}}(r)$ than $r\;=\;10{r}_{{\rm{g}}}$ does (see the third paragraph of Section 4.1). It is, however, possible to compute the RIAF (including ADAF) down to the horizon, if we solve the hydrodynamical Equations (Manmoto 2000; Li et al. 2008) or the magnetohydrodynamical Equations (Punsly et al. 2009; McKinney et al. 2012) in the Kerr spacetime. For instance, ${n}_{{\rm{s}},\mathrm{obs}}$ may increase inward with ∝r⁻² also in $r\lt 6{r}_{{\rm{g}}};$ in this (simplified) case, we would underestimate ${n}_{{\rm{s}},\mathrm{obs}}$ at the gap center ($r\;=\;2{r}_{{\rm{g}}}$) by nine times. An underestimate of ${n}_{{\rm{s}},\mathrm{obs}}$, in turn, incurs an optimistic evaluation of the charge-starvation condition, ${n}_{{\rm{s}},\mathrm{obs}}\lt {n}_{\mathrm{GJ}}$, although the r dependence of B (and hence that of ${n}_{\mathrm{GJ}}$) is non-trivial. If an observation shows that the MeV photon density gives ${n}_{{\rm{s}},\mathrm{obs}}\gt {n}_{\mathrm{GJ}}$, gaps cannot be formed around the BH during the period. To constrain the pair-creating photon density more conclusively, simultaneous observations in TeV and MeV energies are crucial.

As noted in Section 2, the gap center shift outwards if electrons are injected across the inner boundary. In particular, Figure 7 shows that the gap center will shift to $r\gt 6{r}_{{\rm{g}}}$ if the injected current attains $\approx {\rm{\Omega }}B/(2\pi )$. Since a low plasma density is expected around the stagnation surface, it is reasonable to consider a gap formation around it under the existence of such a substantial current injection across the inner boundary.

From the analogy to the pulsar outer-gap model (Hirotani & Shibata 2001b), we can expect that the BH gap electrodynamics is little affected by the change of the gap position. This is because the gap solution is essentially described by the factor $w/{l}_{\mathrm{GJ}}$ (e.g., Equation (21)), where ${l}_{\mathrm{GJ}}$ denotes the length scale in which ${\rho }_{\mathrm{GJ}}$ changes substantially. If the gap is located around the null surface, we can put ${l}_{\mathrm{GJ}}\approx {r}_{{\rm{H}}};$ this dimension analysis would give the similar results as we have derived solving the spatial dependence of ${\rho }_{\mathrm{GJ}}$ explicitly. On the contrary, if the gap is located around the stagnation surface at radius r, we can put ${l}_{\mathrm{GJ}}\approx r$. Thus, if the gap extends enough and $w\sim r$ holds, the potential drop approaches the EMF exerted across the horizon and ${L}_{\mathrm{gap}}$ approaches ${L}_{\mathrm{sd}}$, again in the same manner as pulsars. Thus, the results will not change dramatically if the gap center shifts from the null surface to another place, such as the stagnation surface.

In Section 2, we point out the two representative positions for a gap to be formed: the null surface and the stagnation surface. It is, however, worth noting that more than one gap will not be formed along the same magnetic field lines. This is because the pairs cascaded outside one gap will migrate into the other gap to screen the ${E}_{\parallel }$ there when they polarize. That is, an injection of pairs quenches the gap when one of the charges (e.g., electrons) returns, whereas an injection of a completely charge-separated flow, which consists of only one sign of charges (e.g., positrons), merely shifts the gap position. In the present solutions, the gap around the null surface is not quenching another gap that would arise at another place. The Poisson Equation (5) merely does not allow a gap to be formed around places other than the null surface, if there is no current injection across either the inner or the outer boundaries. In subsequent papers, we will examine a gap formation at another positions (e.g., around the separation surface), considering a current injection across the inner boundary.

In the present paper, we applied our model to IC 310, while LR11 and BT15 compared their models to M87 and SgrA*. Apart from the applied objects, there are important electrodynamical differences among the three works. We discuss this point in this section.

6.3.1. Comparison with LR11

Let us compare the present work with LR11. The main difference appears in two points: the dominant emission process and the gap closure condition.

First, we consider the dominant emission process. To demonstrate the formation of a power-law-like spectrum in VHE, LR11 assumed that the IC process takes place in the Thomson regime and evaluated ${\gamma }_{\mathrm{IC}}$ by

$$\begin{eqnarray}&&{\gamma }_{\mathrm{Thomson}}\;=\;\sqrt{\displaystyle \frac{{{eE}}_{\parallel }}{{\sigma }_{{\rm{T}}}{u}_{{\rm{s}}}}},\end{eqnarray} \tag{ 52 }$$

where ${\sigma }_{{\rm{T}}}$ denotes the Thomson cross section and ${u}_{{\rm{s}}}$ the soft-photon energy density. We should notice here that only the soft photons having energies below ${E}_{{\rm{s}}}\;=\;{m}_{{\rm{e}}}{c}^{2}/\gamma$ contribute in the Thomson regime, unless collisions take place with tiny angles. To compute ${u}_{{\rm{s}}}$, we have to integrate the ADAF photon density up to the energy ${E}_{{\rm{s}}}$. Therefore, if ${E}_{{\rm{s}}}$ appears to be much less than the ADAF peak energy, we obtain ${u}_{{\rm{s}}}\ll {u}_{\mathrm{submm}}$, where ${u}_{\mathrm{submm}}$ denotes the ADAF photon density in submillimeter wavelengths and essentially represents the soft-photon density integrated over all the photon energies.

In the present case, $\gamma \gt 3\times {10}^{9}$ (solid curve in the middle panel of Figure 8) means that only the ADAF photons with energies ${E}_{{\rm{s}}}\lt 2\times {10}^{-4}$ eV contribute in Thomson regime. Since the ADAF spectrum peaks between 10⁻³ eV and 10⁻² eV, we thus obtain ${u}_{{\rm{s}}}\ll {u}_{\mathrm{submm}}$. The thin dotted curve in the middle panel of Figure 8 is computed with this reduced ${u}_{{\rm{s}}}$. On the other hand, LR used ${u}_{\mathrm{submm}}$ to evaluate Equation (52), underestimating ${\gamma }_{\mathrm{IC}}$ such that ${\gamma }_{\mathrm{IC}}\lt {\gamma }_{\mathrm{curv}}$, or equivalently, the the IC process dominates the curvature process.

In short, the leptonic Lorentz factors turned out to be regulated via the curvature process in the present paper, while LR11 considered that they are regulated via the IC process. To compute the IC-limited Lorentz factors, which is essential to examine the closure condition, we employ the Klein–Nishina cross section taking account of the ADAF spectrum, while LR11 employed the Thomson limit using a single parameter ${u}_{\mathrm{submm}}$.

Second, we compare the gap closure condition. LR11 considered a necessary condition for a BH gap to be sustained and imposed that the cascaded pair density should exceed the GJ value outside the gap. They proposed the closure condition, ${{ \mathcal M }}_{\mathrm{LR}}\gt {n}_{\mathrm{pair}}/(| {\rho }_{\mathrm{GJ}}| /e)$, where ${{ \mathcal M }}_{\mathrm{LR}}$ denotes the multiplicity of cascaded pairs from a single primary lepton (their Equation (25)). However, as demonstrated in the bottom panel of Figure 8, the cascaded pair density always exceeds the GJ value (horizontal dotted line). Thus, our solutions automatically satisfy the necessary condition proposed by LR11.

Third, we discuss minor differences. For one thing, we consider the gap arising around the null surface, while LR11 considred the gap arising around the stagnation surface. Nevertheless, it will not incur a qualitative difference as demonstrated in the pulsar outer-gap model (Hirotani & Shibata 2001b). What is more, our results show that the γ-ray luminosity is proportional to ${(w/{r}_{{\rm{g}}})}^{4}$ while LR11 showed it to be proportional to ${(w/{r}_{{\rm{g}}})}^{2};$ the additional ${(w/{r}_{{\rm{g}}})}^{2}$ dependence comes from the fact that the electric potential drop in the gap is proportional to w and that the maximum current is also roughly proportional to w (Equation (19)). For pulsar out-gap models, the gap is geometrically thin in the transverse direction of the magnetic field; thus, the potential drop is proportional to ${({w}_{\perp }/{r}_{{\rm{g}}})}^{2}$, where ${w}_{\perp }$ refers to the trans-magnetic-field thickness of the gap. As a result, the gap luminosity is proportional to ${({w}_{\perp }/{r}_{{\rm{g}}})}^{3}$ (Hirotani 2008) because the total current flowing in the gap is proportional to the gap cross section, which is proportional to ${w}_{\perp }$. However, a BH gap is thin in the longitudinal direction of the magnetic field; thus, the potential drop is proportional to ${(w/{r}_{{\rm{g}}})}^{3}$, which leads to the ${(w/{r}_{{\rm{g}}})}^{4}$ dependence of the gap luminosity (Equation (21)). One final point is that we explicitly computed the photon energy dependence of the pair-creation rate, ICS emissivity, and curvature emissivity, while LR11 adopted a monochromatic approximation for the photon specific intensity.

6.3.2. Comparison with BT15

Let us next compare the electrodynamics of this paper with that of BT15. In the latter paper, they neglected the ${\rho }_{\mathrm{GJ}}$ term in the Poisson equation, their Equation (100). We interpret that they assumed a very large created charge density in the gap, that is, $| \rho | \gg | {\rho }_{\mathrm{GJ}}|$. The reasons are as follows: if the created charge density were sub-GJ, $| {E}_{\parallel }|$ would become less than the vacuum case, leading to $| {E}_{\parallel }| \lt {E}_{\perp }{(w/{l}_{\mathrm{GJ}})}^{2}$, where ${E}_{\perp }\;=\;{{\rm{\Omega }}}_{{\rm{F}}}\varpi B/c$ (i.e., their Equation (1)), ϖ denotes the distance from the rotation axis, w corresponds to Δ in their notation, and ${l}_{\mathrm{GJ}}$ is the distance in which the GJ charge density changes substantially. In the case of BT15, they assume that a gap is located at the stagnation surface, namely between $r\approx 5{r}_{{\rm{g}}}$ and $10{r}_{{\rm{g}}};$ thus, ${l}_{\mathrm{GJ}}\gt 5{r}_{{\rm{g}}}$, which becomes $\approx 5\times {10}^{15}$ cm for M87. It follows that their solution of ${\rm{\Delta }}\approx 4.3\times {10}^{11}$ cm (their Equation (57)) gives ${E}_{\parallel }\approx \times {10}^{-8}{E}_{\perp }$, about 100 million times smaller than their $E\;=\;{E}_{\perp }$ (their Equation (1)), as long as the created charge is sub-GJ as sketched in Figure 7. To overcome difficulty, one must resort on a super-GJ charge density created in the gap, $\rho \gg {\rho }_{\mathrm{GJ}}$. For example, if the real charge density is $\sim {10}^{8}$ times greater than the GJ value, it appears that the Poisson equation might give ${E}_{\parallel }\sim {E}_{\perp }$, that is, their Equation (1), although ${\rm{\Delta }}/{l}_{\mathrm{GJ}}\sim {10}^{-4}$.

If the created charge density were super GJ, it would distribute as the red solid line in the middle panel of Figure 20. However, in this case, $\rho -{\rho }_{\mathrm{GJ}}$ is positive (or negative) in the inner (or outer) part of the gap, leading to a positive ${E}_{\parallel }$, which in turn contradicts the outwardly decreasing ρ. On the contrary, if ρ increases outward, a negative (or positive) $\rho -{\rho }_{\mathrm{GJ}}$ in the inner (or outer) part of the gap, would lead to a negative ${E}_{\parallel }$, which contradicts the outwardly increasing ρ. Thus, we consider that a super-GJ charge creation within the gap is implausible. This is, indeed, the physical reason why all the pulsar gap models treat sub-GJ charge creation within the gap.

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On these grounds, we propose to replace their Equation (1) with $| {E}_{\parallel }| \approx {E}_{\perp }{(w/{l}_{\mathrm{GJ}})}^{2}$, which approximates our Equation (12). In addition, we propose to replace their Equation (17) with our condition ${ \mathcal M }\;=\;1$ (Equation (41)), or with ${{ \mathcal M }}_{\mathrm{LR}}\gt {n}_{\pm }/(| {\rho }_{\mathrm{GJ}}| /e)$ of LR11.

6.3.3. Comparison with HO98

Let us compare the present BH gap model with the pulsar outer-magnetospheric particle accelerator model, that is, the OG model.

First, in a pulsar magnetosphere, the null surface appears because of the convex geometry of a dipole magnetic field. On the other hand, in a BH magnetosphere, a null surface appears due to the frame-dragging effect around a rotating BH. Thus, for the same magnetic field polarity, the sign of ${E}_{\parallel }$ is opposite. In a BH magnetosphere, a pulsar-like null surface could also appear far away from the horizon, in principle. However, unless a strong ring current flows within a certain radius, a dipolar field will not be formed in a BH magnetosphere. Various numerical simulations (Hirose et al. 2004; McKinney 2006; McKinney & Narayan 2007a; Punsly et al. 2009; Tchekhovskoy et al. 2010; McKinney et al. 2012) seems to rule out the formation of such a dipolar-like field geometry, except for the initial-condition-dependent closed poloidal field lines in the equatorial torus imposed far away from the horizon.

Second, in the OG model, a soft-photon field is provided by the cooling NS thermal emission and/or the heated polar-cap thermal emission. The NS surface emissions peak in the X-ray energies. Thus, the curvature 1–10 GeV photons efficiently materialize as pairs within the gap, thereby contributing to the gap closure. For very young pulsars like the Crab pulsar, IC photons could also materialize. However, the thermal IR photon field is much weaker than the thermal X-ray field; thus, the IC photons do not contribute to gap closure in pulsars. In the BH gap model, it is the ADAF that provides the soft-photon field. The ADAF spectrum peaks in submillimeter wavelengths. Thus, the 100 TeV IC photons efficiently materialize as pairs to close the gap. The 0.1–10 TeV curvature photons, on the other hand, do not materialize efficiently because the IR photon number flux is five to six orders of magnitude weaker than the submillimeter one.

Third, if an NS is isolated or is not accreting plasmas from the companion in a binary system, the magnetosphere becomes highly vacuum. The pair density becomes comparable to the GJ value if the space-charge-limited flow is drawn from the NS surface, or becomes much less than the GJ charge value if there is no such a plasma injection. Therefore, a gap is inevitably formed to replenish the charge-starved magnetosphere with a large duty cycle, which is essentially unity. In a BH magnetosphere, on the other hand, ADAF can supply enough charges not only in the lower latitudes but also in the polar funnel via photon–photon pair production. In the case of IC 310, the BH gap is, therefore, expected to be intermittent because the BH gap can be turned on only when the accretion rate becomes less than half of the time-averaged value.

Fourth, an NS magnetosphere is formed by the electric currents inside the NS, whereas a BH magnetosphere is formed by those outside the BH. Thus, the magnetic field energy dominates the plasma energy in an NS magnetosphere, whereas it is at most comparable to the plasma's gravitational binding energy in a BH magnetosphere. Thus, an accreting NS allows accretion along the higher-latitude magnetic field lines that would be open without accretion. This enhanced plasma density quenches an OG, once plasma accretion takes place in a binary system. On the contrary, an accreting BH allows accretion only in the lower latitudes (i.e., near the equator with a certain vertical height), prohibiting plasma penetration into the funnel region because the centrifugal-force barrier prevents plasma accretion toward the rotation axis. Thus, even under plasma accretion, a BH gap can be switched on, provided that the ADAF-supplied pair density (via photon–photon collisions) is small compared to the GJ value.

Fifth and finally, OG distribution is highly non-axisymmetric for an oblique rotator (i.e., a pulsar). This is because ${\rho }_{\mathrm{GJ}}$ and hence the null surface distributes highly non-axisymmetrically, reflecting the three-dimensional magnetic field geometry. However, ${\rho }_{\mathrm{GJ}}$ and hence the gap distributes nearly spherically in a BH magnetosphere, irrespective of the magnetic field geometry (or the magnetic inclination angle), as long as ${{\rm{\Omega }}}_{{\rm{F}}}$ is nearly constant for all the field lines. Therefore, the BH gap solution, which is governed by the ${\rho }_{\mathrm{GJ}}$ distribution, little depends on the magnetic field geometry (e.g., split-monopole or parabolic) because the ${\rho }_{\mathrm{GJ}}$ distribution is essentially determined by the frame-dragging effect near the horizon, instead of the functional form of ${A}_{\varphi }(r,\theta )$, particularly in the higher latitudes (Figure 1).

In addition to the differences pointed out above, there are, of course, similarities between the present BH gap model and the OG model because the basic equations are exactly the same.

First and foremost, the gap width, and hence the γ-ray efficiency, which is defined by the ratio between the γ-ray and spin-down luminosities, increases with decreasing soft-photon flux. This is because the pair-production mean-free path increases with decreasing photon flux. For instance, the OG γ-ray efficiency increases with decreasing NS surface emission, or equivalently with increasing NS age (Hirotani 2013). In the same manner, the BH gap γ-ray efficiency increases with decreasing accretion rate of ADAF.

Next, the superposition of the curvature radiation from different positions produces a nearly power-law spectrum in a limited energy range. In a pulsar magnetosphere, the caustic effect results a compilation of emission from different altitudes with varying ${\rho }_{{\rm{c}}}$ and ${E}_{\parallel }$. However, in a BH magnetosphere, highly tangled, instantaneous magnetic field lines possibly result in a superposition of the curvature emission from different places with varying ${\rho }_{{\rm{c}}}$ and ${E}_{\parallel }$.

In the present paper, we have employed the Newtonian, self-similar, analytical solution of the ADAF (Mahadevan 1997) to specify the background, soft-photon field that illuminates the gap. What is more, for simplicity, we have assumed that the soft-photon field is homogeneous within $10{r}_{{\rm{g}}}$ and its flux reduces by ${r}^{-2}$ law outside this radius, and that the collision angles between these soft photons and the hard γ-rays are isotropic. However, in a realistic BH magnetosphere, position-dependent specific intensity of the soft-photon field is necessary to further quantify the photon–photon absorption as well as the IC scatterings. Thus, the next step may be to incorporate the GR calculations of RIAF. Since the photon distribution is found to be highly inhomogeneous and anisotropic around a rapidly rotating BH (Li et al. 2009), we may be able to constrain the spin of the individual BHs, if we combine the present method, which specifies the emissivity distribution in the magnetosphere, with a sophisticated, numerical RIAF theory.