LEPTON ACCELERATION IN THE VICINITY OF THE EVENT HORIZON: HIGH-ENERGY AND VERY-HIGH-ENERGY EMISSIONS FROM ROTATING BLACK HOLES WITH VARIOUS MASSES

Kouichi Hirotani; Hung-Yi Pu; Lupin Chun-Che Lin; Hsiang-Kuang Chang; Makoto Inoue; Albert K. H. Kong; Satoki Matsushita; Pak-Hin T. Tam

doi:10.3847/1538-4357/833/2/142

1. INTRODUCTION

It is widely accepted that an accreting black hole (BH) of an arbitrary size can produce a pair of relativistic plasma jets, which are often observed in various wavelengths from radio to very-high-energy (VHE) γ-rays. The most plausible mechanism for powering such jets is the extraction of the BH rotational energy through the Blandford–Znajec (BZ) process (Blandford & Znajek 1976). In this electromagnetic process, the magnetic field lines threading the event horizon exert a counter torque on it to spin down the BH, launching Poynting-flux-dominated outflows (Koide et al. 2002). Indeed, general relativistic (GR) magnetohydrodynamic (MHD) models show the existence of nearly steady, collimated, magnetically dominated jets in the polar regions (Hirose et al. 2004; McKinney & Gammie 2004; Tchekhovskoy et al. 2010), whose structures are similar to those in the force-free models (Hawley & Krolik 2006; McKinney & Narayan 2007a, 2007b). Since the centrifugal-force barrier prevents plasma accretion toward the rotation axis, the magnetic energy density dominates the plasmas' rest-mass energy density in these polar funnels.

Even if the axial funnels are almost force-free in this sense, it is the electric current that sustains the electromagnetic power, and it is the charged particles that carry the electric current. That is, even under the assumption of masslessness, it must be the real charged particles that constitute the electric currents. In the direct vicinity of the horizon, causality requires that positive (or negative) charges must flow into the horizon when an electric current flows into (or out from) the horizon. Since accreting plasmas cannot easily penetrate into the funnels, and since they cannot emit sufficient MeV photons that are capable of materializing as electron–positron pairs when the accretion rate is very low, we need a process of plasma supply deep within the magnetosphere somewhere above the horizon.

To contrive a plasma source in the vicinity of the horizon, Beskin et al. (1992) extended the pulsar outer-magnetospheric lepton accelerator model (Cheng et al. 1986a, 1986b) to BH magnetospheres, and proposed the BH gap model. Extending this pioneering work, Hirotani & Okamoto (1998) demonstrated that a sufficient electric current can be supplied via copious pair production around supermassive BHs (SMBHs). However, the predicted γ-ray luminosity was too small to be detected by any instruments, because they assumed a substantial accretion rate (as in quasars), which leads to a very thin gap width along the magnetic field lines.

On these grounds, Neronov & Aharonian (2007) and Levinson & Rieger (2011) revisited the BH gap model and applied it to the central BH of radio galaxy M87 (i.e., M87*) and Srg A*, adopting a much thicker gap width as large as the horizon radius. Their assumption of a thick gap was, indeed, reasonable, because such a low-luminosity active galactic nucleus generally possesses a less dense radiation field in the center and hence a geometrically extended gap slightly above the horizon. Then Broderick & Tchekhovskoy (2015) demonstrated that the two-stream instability does not grow in the ultra-relativistic, counter-streaming pairs in BH gaps. Subsequently, Hirotani & Pu (2016, hereafter HP16) showed that a gap arises around the so-called "null-charge surface," on which the GR Goldreich–Julian (GJ) charge density vanishes, by solving the set of an inhomogeneous part of the Maxwell equations, lepton equations of motion, and the radiative transfer equation. Then they applied their BH gap model to the radio galaxy IC310, whose central BH (i.e., IC310*) is accreting plasmas at much smaller rate than the Eddington rate. They demonstrated that the gap width becomes as large as the horizon radius when the accretion rate is very low, and that the observed VHE flux during the flare cannot be reproduced by their BH gap model, unless the magnetic field becomes much stronger than the equipartition value with the plasma accretions. Noting that GR effects most importantly appear in the formation of the null-charge surface through frame dragging, they evaluated the GJ charge density general-relativistically. However, they neglected the GR effects in any other terms of the basic equations as the first step, because the electromagnetic interaction dominates the gravitational one except for in the direct vicinity of the horizon, and because the photons emitted from the horizon vicinity will not strongly affect the emission spectra due to redshift.

In the present paper, to further quantify the gap model, we include the GR effects in all the basic equations and calculations, extending the method of HP16. In the next section, we examine the detectability of BH gap emissions for various BH masses. Then in Section 3, we describe the background spacetime and derive the GR GJ charge density in a rotating BH magnetosphere. We then formulate the basic equations of gap electrodynamics in Section 4, and investigate their emission properties in Section 5. In the final section, we discuss how to discriminate the gap emission from the jet emission.

2. DETECTABILITY OF GAP EMISSIONS

Since the gap liberates only a portion of the electromagnetic power extracted from a rotating BH, the upper limit of its luminosity can be given by the Blandford–Znajek power (Blandford & Znajek 1976)

$\begin{eqnarray}&&{L}_{\mathrm{BZ}}\approx {10}^{21}{a}_{* }{}^{2}{M}_{1}{}^{2}{B}^{2}\,{{\rm{erg}}{\rm{s}}}^{-1},\end{eqnarray} \tag{ 1 }$

where ${a}_{* }\equiv a/{r}_{{\rm{g}}}$ denotes the dimensionless BH's spin parameter, a ≡ J/(Mc) the spin parameter, J the BH's angular momentum, M the BH mass, c the speed of light, ${r}_{{\rm{g}}}\equiv {{GMc}}^{-2}$ the gravitational radius, G the gravitational constant, B the magnetic field strength in Gauss, and ${M}_{1}\equiv M/(10\,{M}_{\odot })$ .

In the present paper, unless explicitly mentioned, we assume that the magnetic buoyancy balances disk gravity and evaluate B with the equipartition value. Near the horizon, we obtain (Bisnovatyi-Kogan & Ruzmaikin 1974; Ghosh et al. 1977; Narayan et al. 2003; Levinson & Rieger 2011)

$\begin{eqnarray}&&{B}_{\mathrm{eq}}\approx 4\times {10}^{8}{\dot{m}}^{1/2}{M}_{1}{}^{-1/2}\,{\rm{G}},\end{eqnarray} \tag{ 2 }$

where $\dot{m}$ refers to the dimensionless accretion rate near the horizon and is defined as $\dot{m}\equiv \dot{M}/{\dot{M}}_{\mathrm{Edd}};$ $\dot{M}$ denotes the mass accretion rate, ${\dot{M}}_{\mathrm{Edd}}$ ≡ ${L}_{\mathrm{Edd}}/\eta {c}^{2}$ , L_Edd = 1.25 × 10³⁹ M₁ erg s⁻¹, and η ≈ 0.1 (McKinney et al. 2012). Substituting B = B_eq into Equation (1), we obtain

$\begin{eqnarray}&&{L}_{\mathrm{BZ}}\approx 1.7\times {10}^{38}{a}_{* }{}^{2}\dot{m}{M}_{1}\,{{\rm{erg}}{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 3 }$

The magnetosphere becomes force-free, that is, an electric field does not arise along the magnetic field lines, if the pair density exceeds the GJ value, ${N}_{\mathrm{GJ}}\sim {{\rm{\Omega }}}_{{\rm{F}}}B/(2\pi {ce})$ , where Ω_F denotes the angular frequency of the magnetic field and e the magnitude of the charge on the electron. In the vicinity of an accreting BH, a plasma accretion becomes radiatively inefficient when the accretion rate typically reduces to $\dot{m}\lt {10}^{-2.5}$ (Ichimaru 1979; Narayan & Yi 1994, 1995). From such a radiatively inefficient accretion flow (RIAF), MeV photons are emitted via a free–free process, colliding with each other to materialize as electron–positron pairs in the magnetosphere. If the number density, N_±, of such created pairs becomes less than N_GJ, charges cannot completely screen an electric field, E_∥, along the magnetic field line; that is, a gap (i.e., a charge-starved region) appears. The RIAF theory gives N_± as a function of M and $\dot{m}$ (Levinson & Rieger 2011). Thus, putting N_± < N_GJ, we obtain a condition for a gap to appear,

$\begin{eqnarray}&&\dot{m}\lt {\dot{m}}_{\mathrm{up}}\equiv 3.1\times {10}^{-3}{M}_{1}{}^{-1/7}.\end{eqnarray} \tag{ 4 }$

In Figure 1, we plot this upper limit, ${\dot{m}}_{\mathrm{up}}={\dot{m}}_{\mathrm{up}}(M)$ as the thick solid line. The dotted lines denote L_BZ (Equation (3)) as labeled. The crossing of the solid line and a dotted line gives the Blandford–Znajek power at each BH mass. For example, the thick solid line and the dotted line labeled L_BZ = 10⁴⁰ erg s⁻¹ cross at $M\sim {10}^{6}\,{M}_{\odot };$ thus, we find that a SMBH with M ∼ 10⁶ M_⊙ has a gap whose luminosity can attain up to 10⁴⁰ erg s⁻¹.

To consider the case of an efficient energy extraction from a BH, we assume ${a}_{* }=0.9$ unless explicitly mentioned. Substituting $\dot{m}={\dot{m}}_{\mathrm{up}}$ into Equation (3), we then obtain the maximum gap luminosity

$\begin{eqnarray}&&{L}_{\mathrm{BZ}}\approx 5.2\times {10}^{35}{a}_{* }{}^{2}{M}_{1}{}^{6/7}\,{\text{erg s}}^{-1}.\end{eqnarray} \tag{ 5 }$

Assuming that 100% of this power is converted into radiation, we obtain the upper limit of its flux at Earth, F_BZ = L_BZ/4πd², where d is the distance to the BH.

For stellar-mass BHs, we obtain the flux upper limit,

$\begin{eqnarray}&&{F}_{\mathrm{BZ}}=\displaystyle \frac{{L}_{\mathrm{BZ}}}{4\pi {d}^{2}}=4.1\times {10}^{-9}{a}_{* }{}^{2}{M}_{1}{}^{6/7}{\left(\displaystyle \frac{d}{\mathrm{kpc}}\right)}^{-2}{\text{erg s}}^{-1}\,{{\rm{cm}}}^{-2}.\end{eqnarray} \tag{ 6 }$

As will be seen in Section 5.1, the gap emission spectrum peaks between GeV and 10 GeV for $M\sim 10\,{M}_{\odot }$ . Thus, if a small portion of this BZ power (e.g., 0.1%) is dissipated in the gap, we can expect a large time-averaged high-energy (HE) flux (e.g., >10⁻¹¹ erg s⁻¹) that is detectable with Fermi/LAT. There is another component, formed by the inverse-Compton (IC) scatterings, which appears in VHE. This component may be detectable with ground-based imaging atmospheric Cherenkov telescopes (IACTs). Note that Equation (6) merely gives the upper limit, and that the actual photon flux can be obtained when we solve the gap electrodynamics from the basic equations. Will will examine this issue in Section 5.1.

For intermediate-mass BHs, we obtain the maximum flux of

$\begin{eqnarray}&&{F}_{\mathrm{BZ}}=2.1\times {10}^{-9}{a}_{* }{}^{2}{M}_{3}{}^{6/7}{\left(\displaystyle \frac{d}{10\mathrm{kpc}}\right)}^{-2}{\text{erg s}}^{-1}\,{{\rm{cm}}}^{-2},\end{eqnarray} \tag{ 7 }$

where M₃ ≡ M/(10³ M_⊙). The spectrum has two peaks: the curvature photons peak in 1–10 GeV and the IC ones above TeV. Both spectral components are potentially detectable in HE and VHE if they are located within our galaxy. We will examine this possibility in Section 5.2.

For supermassive BHs, we obtain the maximum flux of

$\begin{eqnarray}&&{F}_{\mathrm{BZ}}=3.0\times {10}^{-10}{a}_{* }{}^{2}{M}_{9}{}^{6/7}{\left(\displaystyle \frac{d}{10\mathrm{Mpc}}\right)}^{-2}\,{\text{erg s}}^{-1}\,{{\rm{cm}}}^{-2},\end{eqnarray} \tag{ 8 }$

where ${M}_{9}\equiv M/({10}^{9}\,{M}_{\odot })$ . The IC component, which appears in VHE, may be detectable with ground-based IACTs. We will examine this possibility in Section 5.3.

3. BACKGROUND GEOMETRY AND THE NULL CHARGE SURFACE

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)

$\begin{eqnarray}&&{{ds}}^{2}={g}_{{tt}}{{dt}}^{2}+2{g}_{t\varphi }{dtd}\varphi +{g}_{\varphi \varphi }d{\varphi }^{2}+{g}_{{rr}}{{dr}}^{2}+{g}_{\theta \theta }d{\theta }^{2},\end{eqnarray} \tag{ 9 }$

where

$\begin{eqnarray}&&{g}_{{tt}}\equiv -\displaystyle \frac{{\rm{\Delta }}-{a}^{2}{\sin }^{2}\theta }{{\rm{\Sigma }}}{c}^{2},\qquad {g}_{t\varphi }\equiv -\displaystyle \frac{2({GM}/{c}^{2}){ar}{\sin }^{2}\theta }{{\rm{\Sigma }}}c,\end{eqnarray} \tag{ 10 }$

$\begin{eqnarray}&&{g}_{\varphi \varphi }\equiv \displaystyle \frac{A{\sin }^{2}\theta }{{\rm{\Sigma }}},\qquad {g}_{{rr}}\equiv \displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}},\qquad {g}_{\theta \theta }\equiv {\rm{\Sigma }};\end{eqnarray} \tag{ 11 }$

${\rm{\Delta }}\equiv {r}^{2}-2({GM}/{c}^{2})r+{a}^{2}$ , ${\rm{\Sigma }}\equiv {r}^{2}$ + ${a}^{2}{\cos }^{2}\theta$ , $A\,\equiv {({r}^{2}+{a}^{2})}^{2}\,-$ ${\rm{\Delta }}{a}^{2}{\sin }^{2}\theta$ . 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. If the BH is extremely rotating (i.e., if $a\to {r}_{{\rm{g}}}$ ) for instance, we obtain ${r}_{{\rm{H}}}\to {r}_{{\rm{g}}}$ .

In a stationary and axisymmetric spacetime (as described by Equation (9)), Gauss's law becomes

$\begin{eqnarray}&&{{\rm{\nabla }}}_{\mu }{F}^{t\mu }=\displaystyle \frac{1}{\sqrt{-g}}{\partial }_{\mu }\left[\displaystyle \frac{\sqrt{-g}}{{\rho }_{{\rm{w}}}^{2}}{g}^{\mu \nu }(-{g}_{\varphi \varphi }{F}_{t\nu }+{g}_{t\varphi }{F}_{\varphi \nu })\right]=\displaystyle \frac{4\pi }{{c}^{2}}\rho ,\end{eqnarray} \tag{ 12 }$

where ∇ denotes the covariant derivative, the Greek indices run over t, r, θ, φ; $\sqrt{-g}=\sqrt{{g}_{{rr}}{g}_{\theta \theta }{\rho }_{{\rm{w}}}^{2}}=c{\rm{\Sigma }}\sin \theta$ and ${\rho }_{{\rm{w}}}^{2}\equiv {g}_{t\varphi }^{2}-{g}_{{tt}}{g}_{\varphi \varphi }={c}^{2}{\rm{\Delta }}{\sin }^{2}\theta$ , ρ is the real charge density. The electromagnetic fields are observed by an observer that is static with respect to asymptotic infinity, and are given by (Camenzind 1986a, 1986b) ${E}_{r}={F}_{{rt}},{E}_{\theta }={F}_{\theta t},{E}_{\varphi }={F}_{\varphi t}$ , ${B}^{r}=({g}_{{tt}}+{g}_{t\varphi }{\rm{\Omega }})$ ${F}_{\theta \varphi }/\sqrt{-g}$ , ${B}^{\theta }=({g}_{{tt}}+{g}_{t\varphi }{\rm{\Omega }}){F}_{\varphi r}/\sqrt{-g}$ , ${B}_{\varphi }=-{\rho }_{{\rm{w}}}^{2}{F}_{r\theta }/\sqrt{-g}$ , where ${F}_{\mu \nu }\equiv {\partial }_{\mu }$ ${A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ and A_μ denotes the vector potential.

In this paper, we assume that the electromagnetic fields (i.e., all components of the Faraday tensor, ${F}_{\mu \nu }$ ) depend on t and φ through $\varphi -{{\rm{\Omega }}}_{{\rm{F}}}t$ . In this case, we can introduce the non-corotational potential Φ such that

$\begin{eqnarray}&&{F}_{\mu t}+{{\rm{\Omega }}}_{{\rm{F}}}{F}_{\mu \varphi }=-{\partial }_{\mu }{\rm{\Phi }}(r,\theta ,\varphi -{{\rm{\Omega }}}_{{\rm{F}}}t).\end{eqnarray} \tag{ 13 }$

If ${F}_{{At}}+{{\rm{\Omega }}}_{{\rm{F}}}{F}_{A\varphi }=0$ holds for A = r and θ, ${{\rm{\Omega }}}_{{\rm{F}}}\equiv {F}_{{tr}}/{F}_{r\varphi }={F}_{t\theta }/{F}_{\theta \varphi }$ is conserved along the field line. However, in a particle acceleration region, ${F}_{{At}}+{{\rm{\Omega }}}_{{\rm{F}}}{F}_{A\varphi }$ deviates from 0 and the magnetic field does not rigidly rotate. The deviation from rigid rotation is expressed in terms of Φ, which gives the strength of the acceleration electric field measured by an observer that is static to asymptotic infinity as

$\begin{eqnarray}&&{E}_{\parallel }\equiv \displaystyle \frac{{\boldsymbol{B}}}{B}\cdot {\boldsymbol{E}}=\displaystyle \frac{{B}^{i}}{B}({F}_{{it}}+{{\rm{\Omega }}}_{{\rm{F}}}{F}_{i\varphi })=\displaystyle \frac{{\boldsymbol{B}}}{B}\cdot (-{\rm{\nabla }}{\rm{\Phi }}),\end{eqnarray} \tag{ 14 }$

where the Latin index i runs over spatial coordinates r, θ, φ. Note that ${B}^{i}{F}_{i\varphi }=0$ .

Substituting Equation (13) into (12), we obtain the Poisson equation for the non-corotational potential (Hirotani 2006),

$\begin{eqnarray}&&-\displaystyle \frac{{c}^{2}}{\sqrt{-g}}{\partial }_{\mu }\left(\displaystyle \frac{\sqrt{-g}}{{\rho }_{{\rm{w}}}^{2}}{g}^{\mu \nu }{g}_{\varphi \varphi }{\partial }_{\nu }{\rm{\Phi }}\right)=4\pi (\rho -{\rho }_{\mathrm{GJ}}),\end{eqnarray} \tag{ 15 }$

where the GR Goldreich–Julian charge density is defined as

$\begin{eqnarray}&&{\rho }_{\mathrm{GJ}}\equiv \displaystyle \frac{{c}^{2}}{4\pi \sqrt{-g}}{\partial }_{\mu }\left[\displaystyle \frac{\sqrt{-g}}{{\rho }_{{\rm{w}}}^{2}}{g}^{\mu \nu }{g}_{\varphi \varphi }({{\rm{\Omega }}}_{{\rm{F}}}-\omega ){F}_{\varphi \nu }\right].\end{eqnarray} \tag{ 16 }$

In the limit $r\gg {r}_{{\rm{g}}}$ , Equation (16) reduces to the ordinary, special-relativistic expression (Goldreich & Julian 1969; Mestel 1971),

$\begin{eqnarray}&&{\rho }_{\mathrm{GJ}}\equiv -\displaystyle \frac{{\boldsymbol{\Omega }}\cdot {\boldsymbol{B}}}{2\pi c}+\displaystyle \frac{({\boldsymbol{\Omega }}\times {\boldsymbol{r}})\cdot ({\rm{\nabla }}\times {\boldsymbol{B}})}{4\pi c}.\end{eqnarray} \tag{ 17 }$

Therefore, the corrections due to magnetospheric currents, which are expressed by the second term of Equation (17), are included in Equation (16).

In addition to ${F}_{\mu \nu }$ and Φ, ρ may also depend on t and φ only through $\varphi -{{\rm{\Omega }}}_{{\rm{F}}}t$ . In this case, Equation (15) gives a "stationary" gap solution in the "corotational" frame, in the sense that Φ and ρ are a function of r, θ, and $\varphi -{{\rm{\Omega }}}_{{\rm{F}}}t$ . Note that such stationary solutions are valid not only between the two light surfaces (i.e., where k₀ ≡ −g_tt − 2g_tφΩ_F − g_φφ ${{\rm{\Omega }}}_{{\rm{F}}}^{2}\gt 0$ ), but also inside the inner light surface and outside the outer light surface (i.e., where k₀ < 0; Bogovalov 1999; Pètri & Kirk 2005; Pètri 2013).

Equation (15) shows that ${E}_{\parallel }$ is exerted along ${\boldsymbol{B}}$ if ρ deviates from ${\rho }_{\mathrm{GJ}}$ in any region. In the close vicinity of the inner and outer boundaries, we can neglect the trans-field derivatives to obtain ${\partial }_{r}{E}_{\parallel }\propto \rho -{\rho }_{\mathrm{GJ}}$ . Thus, ${\partial }_{r}{E}_{\parallel }$ , and hence $\rho -{\rho }_{\mathrm{GJ}}$ must have different signs at the two boundaries so that the gap may be closed. In a nearly vacuum gap, ρ ≈ 0 shows that ${\rho }_{\mathrm{GJ}}$ should change sign within the gap. Therefore, a gap should appear around the null-charge surface, where ${\rho }_{\mathrm{GJ}}$ vanishes. The null surface is, therefore, a natural place for a particle accelerator (i.e., a gap) to arise, in the same way as pulsar vacuum gap models (Holloway 1973; Chiang & Romani 1992; Romani 1996; Cheng et al. 2000).

It should be noted that the null surface appears near the place where Ω_F coincides with the spacetime dragging angular frequency, ω (Beskin et al. 1992). The deviation of the null surface from this $\omega (r,\theta )={{\rm{\Omega }}}_{{\rm{F}}}$ surface is, indeed, small, as Figure 1 of Hirotani & Okamoto (1998) indicates. Since ω matches Ω_F only near the horizon, the null surface, and hence the gap, generally appears within one or two gravitational radii above the horizon, irrespective of the BH mass.

4. MAGNETOSPHERIC LEPTON ACCELERATOR NEAR THE HORIZON

In this section, we formulate the BH gap electrodynamics, extending the method described in Hirotani (2013) and HP16. Throughout this paper, we assume an aligned rotator in the sense that the magnetic axis coincides with the rotational axis of the BH, and seek an axisymmetric solution.

4.1. Magnetic Field Structure

As described in HP16, a stationary BH gap is formed around the null surface, as long as the injected current density across the inner or outer boundaries is much smaller than the GJ value. Since the null surface is formed by the frame-dragging effect, the gap electrodynamics is essentially governed by the frame-dragging effect rather than the magnetic field configurations. This forms a striking contrast to the pulsar outer-magnetospheric gap model, in which the null surface is formed by the convex geometry of the poloidal magnetic field lines. Thus, in a BH magnetosphere, the gap position and its spatial extent, as well as the exerted ${E}_{\parallel }={E}_{\parallel }(s)$ in the gap, depend little on the magnetic flux function, ${\rm{\Psi }}={A}_{\varphi }$ . We thus assume a radial magnetic field on the poloidal plane, ${\rm{\Psi }}={\rm{\Psi }}(\theta )$ .

Because of axial symmetry, the gap electrodynamic structure can be described in the poloidal plane. For simplicity, we assume that the photons propagate radially in this 2D plane, which is justified if the photons have negligible angular momenta. Provided that the drift motion (e.g., due to toroidal radiation drag in a radial magnetic field, or due to radial gravity in a toroidal magnetic field) is small in the meridional direction, charged particles roughly move along the magnetic field lines in the poloidal plane. In this case, due to relativistic beaming, ultra-relativistic particles emit photons along the instantaneous magnetic field lines; thus, as long as the poloidal magnetic field is radial, photons propagate on the same magnetic flux surface, ${A}_{\varphi }={A}_{\varphi }(\theta )=\mathrm{constant}$ . As a result, we can solve the particle equations of motion and the radiative transfer equation along individual radial poloidal magnetic field lines separately.

Even when the magnetic field lines are radial in the poloidal plane, there exists a toroidal magnetic component, B_φ, due to retardation, magnetospheric currents, and a frame dragging. It is, however, out of the scope of this paper to restrict the functional form of ${B}_{\varphi }(r,\theta )$ , taking account of such effects. In addition, B_φ does not affect ${\rho }_{\mathrm{GJ}}$ or the propagation direction of particles and photons in the poloidal plane. Thus, we do not specify B_φ. Accordingly, instead of computing the curvature radius, R_c, of the leptons from their 3D motion in the rotating magnetosphere, we parameterize R_c when we calculate the curvature emission. Although R_c affects the spectral shape of curvature emission, it affects the total luminosity little, because the latter is essentially determined by the potential drop within the gap, and because the potential drop is determined by the gap width, which is in turn predominantly determined by the pair production rate of the IC-emitted, VHE photons (not the curvature-emitted, lower energy photons). On these grounds, we adopt R_c = r_g in the present paper, leaving B_φ unconstrained.

4.2. Gap Electrodynamics

In the same way as HP16, we solve the stationary gap solution from the set of the Poisson equation for Φ, the equations of motion for electrons and positrons, and the radiative transfer equation for the emitted photons.

4.2.1. Poisson Equation

To solve the radial dependence of Φ in the Poisson Equation (15), we introduce the following dimensionless tortoise coordinate, η_*,

$\begin{eqnarray}&&\displaystyle \frac{d{\eta }_{* }}{{dr}}=\displaystyle \frac{{r}^{2}+{a}^{2}}{{\rm{\Delta }}}\displaystyle \frac{1}{{r}_{{\rm{g}}}}.\end{eqnarray} \tag{ 18 }$

In this coordinate, the horizon corresponds to the "inward infinity," ${\eta }_{* }=-\infty$ . In this paper, we set ${\eta }_{* }=r/{r}_{{\rm{g}}}$ at $r={r}_{\max }\equiv 25{r}_{{\rm{g}}}$ , where the value of r_max can be chosen arbitrarily and does not affect the results in any way. The distribution of ${\eta }_{* }$ is depicted as a function of r/r_g in Figure 2. Note that the relationship between ${\eta }_{* }$ and r does not depend on the colatitude, θ.

**Figure 2.** Tortoise coordinate vs. Boyer–Lindquist radial coordinate. Both axes are normalized by the gravitational radius,r_g. The BH spin parameter is adopted to be ${a}_{* }=0.90$ . The vertical dashed line represents the horizon position.
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**Figure 2.** Tortoise coordinate vs. Boyer–Lindquist radial coordinate. Both axes are normalized by the gravitational radius,r_g. The BH spin parameter is adopted to be ${a}_{* }=0.90$ . The vertical dashed line represents the horizon position.
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Since the gap is located near the horizon, we take the limit ${\rm{\Delta }}\ll {r}_{{\rm{g}}}{}^{2}$ . Assuming that Φ does not depend on $\varphi -{{\rm{\Omega }}}_{{\rm{F}}}t$ , that is, ${\rm{\Phi }}={\rm{\Phi }}(r,\theta )$ , we can recast the Poisson Equation (15) into the two-dimensional form,

$\begin{eqnarray}&&\quad \quad -{\left(\displaystyle \frac{{r}^{2}+{a}^{2}}{{\rm{\Delta }}}\right)}^{2}\displaystyle \frac{{\partial }^{2}\tilde{{\rm{\Phi }}}}{\partial {\eta }_{* }{}^{2}}+\displaystyle \frac{2(r-{r}_{{\rm{g}}})({r}^{2}+{a}^{2})}{{{\rm{\Delta }}}^{2}/{r}_{{\rm{g}}}}\displaystyle \frac{\partial \tilde{{\rm{\Phi }}}}{\partial {\eta }_{* }}\\ &&\quad \quad -\displaystyle \frac{{r}_{{\rm{g}}}{}^{2}}{{\rm{\Delta }}}\displaystyle \frac{{\rm{\Sigma }}}{\sin \theta }\displaystyle \frac{\partial }{\partial \theta }\left(\displaystyle \frac{\sin \theta }{{\rm{\Sigma }}}\displaystyle \frac{\partial \tilde{{\rm{\Phi }}}}{\partial \theta }\right)\\ &&\quad ={\left(\displaystyle \frac{{\rm{\Sigma }}}{{r}^{2}+{a}^{2}}\right)}^{2}({n}_{+}-{n}_{-}-{n}_{\mathrm{GJ}}),\end{eqnarray} \tag{ 19 }$

where

$\begin{eqnarray}&&\tilde{{\rm{\Phi }}}({\eta }_{* })\equiv \displaystyle \frac{c}{2{{\rm{\Omega }}}_{{\rm{F}}}{{Br}}_{{\rm{g}}}{}^{2}}{\rm{\Phi }}(r)\end{eqnarray} \tag{ 20 }$

denotes the dimensionless non-corotational potential. Dimensionless lepton densities per magnetic flux tube are defined by

$\begin{eqnarray}&&{n}_{\pm }\equiv \displaystyle \frac{2\pi {ce}}{{{\rm{\Omega }}}_{{\rm{F}}}B}{N}_{\pm },\end{eqnarray} \tag{ 21 }$

where the number densities of positrons and electrons, ${N}_{+}$ and ${N}_{-}$ , are computed from the pair production rate at each position (Hirotani & Okamoto 1998; Hirotani & Shibata 1999a, 1999b). Dimensionless GJ charge density per magnetic flux tube is defined by

$\begin{eqnarray}&&{n}_{\mathrm{GJ}}\equiv \displaystyle \frac{2\pi c}{{{\rm{\Omega }}}_{{\rm{F}}}B}{\rho }_{\mathrm{GJ}}.\end{eqnarray} \tag{ 22 }$

For a radial poloidal magnetic field, ${\rm{\Psi }}={\rm{\Psi }}(\theta )$ , we can compute the acceleration electric field by

$\begin{eqnarray}&&{E}_{\parallel }\equiv -\displaystyle \frac{\partial {\rm{\Phi }}}{\partial r}=-{r}_{{\rm{g}}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{F}}}B}{c}\displaystyle \frac{{r}^{2}+{a}^{2}}{{\rm{\Delta }}}\displaystyle \frac{\partial \tilde{{\rm{\Phi }}}}{\partial {\eta }_{* }}.\end{eqnarray} \tag{ 23 }$

Without loss of any generality, we can assume ${F}_{\theta \varphi }\gt 0$ in the northern hemisphere. In this case, a negative ${E}_{\parallel }$ arises in the gap, which is consistent with the direction of the global current flow pattern.

Equation (16) shows that ${\rho }_{\mathrm{GJ}}$ is essentially determined by B^r, rather than B^θ, near the horizon. In a stationary and axisymmetric magnetosphere, Equation (16) becomes

$\begin{eqnarray}&&{\rho }_{\mathrm{GJ}}\propto {\partial }_{r}({{GF}}_{\varphi r})+\displaystyle \frac{1}{{\rm{\Delta }}}{\partial }_{\theta }({{GF}}_{\varphi \theta }),\end{eqnarray} \tag{ 24 }$

where $G\equiv (A\sin \theta /{\rm{\Sigma }})({{\rm{\Omega }}}_{{\rm{F}}}-\omega )$ . Since G, ${F}_{\varphi r}$ , and ${F}_{\varphi \theta }$ are well-behaved at the horizon, we find that the second term dominates the first one at ${\rm{\Delta }}\to 0$ . Therefore, although the null surface itself is formed by the frame-dragging effect, the radial component of the magnetic field essentially determines ${\rho }_{\mathrm{GJ}}$ near the horizon owing to the redshift effect.

4.2.2. Leptonic Densities

We next consider ${n}_{-}$ and ${n}_{+}$ in Equation (19). Because of E_∥ < 0, electrons are accelerated outwards, and positrons inwards. As a result, as long as there is no current injection across either outer or inner boundaries, charge density, n₊ −n₋, becomes negative (or positive) at the outer (or inner) boundary. In a stationary gap, ${E}_{\parallel }$ should not change sign in it. In a vacuum gap, a positive (or a negative) $-{n}_{\mathrm{GJ}}$ near the outer (or inner) boundary makes ${\partial }_{r}{E}_{\parallel }\gt 0$ (or ${\partial }_{r}{E}_{\parallel }\lt 0$ ), thereby closing the gap. In a non-vacuum gap, the right-hand side of Equation (19) should become positive (or negative) near the outer (or inner) boundary so that the gap may be closed. Therefore, $| {n}_{+}-{n}_{-}|$ should not exceed $| {n}_{\mathrm{GJ}}|$ at either boundary. At the outer boundary, for instance, we can put

$\begin{eqnarray}&&({n}_{+}-{n}_{-}){| }_{r={r}_{2}}=-{n}_{-}(r={r}_{2})={{jn}}_{\mathrm{GJ}}(r={r}_{2}),\end{eqnarray} \tag{ 25 }$

where the dimensionless parameter j should be in the range, $0\leqslant j\leqslant 1$ , so that the gap solution may be stationary. Since ${F}_{\theta \varphi }\gt 0$ is assumed, it is enough to consider a positive j. If j = 1, there is no surface charge at the outer boundary. However, if j < 1, the surface charge results in a jump of ${\partial }_{r}{E}_{\parallel }$ at the outer boundary. That is, the parameter j specifies the strength of ${\partial }_{r}{E}_{\parallel }$ at the outer boundary. Thus, the inner boundary position, r = r₁, is determined as a free boundary problem by this additional constraint, j. The outer boundary position, r = r₂, or equivalently the gap width w = r₂ − r₁, is constrained by the gap closure condition (Section 4.2.6).

It is noteworthy that the charge conservation ensures that the dimensionless current density (per magnetic flux tube), ${J}_{{\rm{c}}}\equiv -{n}_{+}-{n}_{-}$ conserves along the flowline. At the outer boundary, we obtain

$\begin{eqnarray}&&{J}_{{\rm{c}}}=-{n}_{-}(r={r}_{2})={{jn}}_{\mathrm{GJ}}(r={r}_{2}).\end{eqnarray} \tag{ 26 }$

Thus, j specifies not only ${\partial }_{r}{E}_{\parallel }$ at the outer boundary, but also the conserved current density, J_c.

In general, under a given electro-motive force exerted in the ergosphere, J_c should be constrained by the global current flow pattern, which includes an electric load at the large distances where the force-free approximation breaks down and the trans-magnetic-field current gives rise to the outward acceleration of charged particles by Lorentz forces (thereby converting the Poynting flux into particle kinetic energies). However, we will not go deep into the determination of J_c in this paper, because we are concerned with the acceleration processes near the horizon, not the global current closure issue. Note that w (or r₂) is essentially determined by $\dot{m};$ thus, j and $\dot{m}$ give the actual current density $({{\rm{\Omega }}}_{{\rm{F}}}B/2\pi ){J}_{{\rm{c}}}$ , where B should be evaluated at each position. On these grounds, instead of determining J_c by a global requirement, we treat j as a free parameter in the present paper.

It may be worth mentioning, in passing, that we may not have to consider a time-dependent solution, which may be obtained when j > 1 as in pulsar polar cap models (Harding et al. 1978; Daugherty & Harding 1982; Dermer & Sturner 1994). In the polar cap model, the absence of the null surface results in a non-stationary gap solution (Timokhin 2010; Timokhin & Arons 2013; Timokhin & Harding 2015). However, in the pulsar outer-gap models (Zhang & Cheng 1997; Takata et al. 2006; Romani & Watters 2010; Wang et al. 2011; Hirotani 2015) or in the present BH gap model, the existence of the null surface leads to the formation of a stationary gap around this surface. We thus adopt 0 ≤ j ≤ 1 and consider a stationary gap solution.

4.2.3. Particle Motion

Let us describe the motion of electrons and positrons. For simplicity, we assume that the distribution functions of electrons and positrons are mono-energetic. We evaluate the Lorentz factors of electrons (or positrons) by the motion of a test particle injected across the inner (or the outer) boundary of the gap. For example, an injected test electron is accelerated by a negative E_∥ outwards and loses the kinetic energy via curvature and IC processes. The former, curvature radiation rate (per particle), is computed by the standard synchrotron emission formula with the gyration radius replaced with R_c (Rybicki & Lightman 1979). The latter, IC radiation rate, is computed by multiplying the scattering probability (per unit time) and the scattered photon energy. Thus, the particle Lorentz factor saturates at the curvature- or IC-limited value, whichever is smaller.

4.2.4. Radiative Transfer Equation

Throughout this paper, we assume that all photons are emitted with vanishing angular momenta. In this case, photons propagate on a constant-θ surface; thus, the radiative transfer equation is solved one-dimensionally along the radial magnetic field lines on the poloidal plane. These primary leptons emit photons via curvature and IC processes both inside and outside the gap. For the details of how to compute the emissivities of curvature and IC processes, see Sections 4.2 and 4.3 of HP16.

Some portions of the photons are emitted above 10 TeV via IC process. A significant fraction of such hard γ-rays are absorbed, colliding with the RIAF soft photons. If such collisions take place within the gap, the created electrons and positrons polarize to be accelerated in opposite directions, becoming the primary leptons. If the collisions take place outside the gap, the created, secondary pairs migrate along the magnetic filed lines to emit photons via IC and synchrotron processes. Some of such secondary IC photons are absorbed again to materialize as tertiary pairs, which emit tertiary photons via synchrotron and IC processes, eventually cascading into higher generations. As a representative model of the RIAF, we adopt the analytic solution of the advection-dominated accretion flow (ADAF) obtained by Mahadevan (1997).

We assume that the ADAF soft photon specific intensity is isotropic in the zero angular momentum observer (ZAMO). This assumption simplifies the calculations of photon–photon pair production and IC scatterings, because photons are assumed to be emitted by leptons with vanishing angular momenta. To calculate the flux of ADAF photons, we assume that their number density is homogeneous and becomes ${L}_{\mathrm{ADAF}}/(4\pi {R}_{\min }^{2}c)$ within r < R_min, where ${L}_{\mathrm{ADAF}}$ denotes the ADAF luminosity given by Mahadevan (1997). We assume that the ADAF luminosity becomes L_ADAF at r = R_min. Outside this radius, r > R_min, we assume that their number density decreases by r⁻² law. This treatment may be justified, because the submillimeter-IR photons, which most effectively work both for pair production and IC scatterings, are emitted from the innermost region of the ADAF.

4.2.5. Boundary Conditions

We solve the gap in the 2D poloidal plane. We assume a reflection symmetry with respect to the magnetic axis. Thus, we put ${\partial }_{\theta }\tilde{{\rm{\Phi }}}=0$ at θ = 0. We assume that the polar funnel is bounded at a fixed colatitude, θ = θ_max and impose that this lower-latitude boundary is equi-potential and put $\tilde{{\rm{\Phi }}}=0$ at $\theta ={\theta }_{\max }$ .

Both the outer and inner boundaries are treated as free boundaries. Their positions are determined by two conditions: the value of j along each magnetic field line (specified by Ψ), and the gap closure condition (to be described in Section 4.2.6). For simplicity, we assume that j is constant for Ψ. At the outer boundary, ${\partial }_{r}{E}_{\parallel }=-{\partial }_{r}{}^{2}{\rm{\Phi }}$ is specified by j. At the inner boundary, we impose ${E}_{\parallel }=-{\partial }_{r}{\rm{\Phi }}=0$ . We assume that electrons, positrons or photons are not injected across either the outer or the inner boundaries.

4.2.6. Gap Closure Condition

The set of Poisson and radiative-transfer equations are solved together with the terminal Lorentz factor γ and the n_± obtained by the local pair production rate. Unlike HP16, we discard the reflection symmetry (along radial magnetic field lines) with respect to the null-charge surface, and explicitly consider the asymmetric distribution of ${E}_{\parallel }$ , γ and n_±, and the photon specific intensity at each point r. Accordingly, the gap closure condition should be modified as ${{ \mathcal M }}_{\mathrm{in}}{{ \mathcal M }}_{\mathrm{out}}=1$ , where ${{ \mathcal M }}_{\mathrm{in}}$ and ${{ \mathcal M }}_{\mathrm{out}}$ denote the multiplicity (Equation (41) of HP16) associated with the in-going and out-going leptons, respectively. See Hirotani (2013) for a detailed treatment of the asymmetric multiplicities, ${{ \mathcal M }}_{\mathrm{in}}\ne {{ \mathcal M }}_{\mathrm{out}}$ .

5. GAP SOLUTIONS

In Section 2, we examined the upper limits of the BH gap luminosity, imposing the charge-starvation condition (Equation (4)) that requires the ADAF to be less luminous so that their MeV photons may not produce pairs above the GJ density. In this section, solving the gap electrodynamics by the method described in Section 4 for various BH masses, we demonstrate that the gap becomes most luminous when the gap longitudinal width becomes much greater than the horizon radius, and that there is a lower-limit accretion rate below which a stationary pair-production cascade cannot be maintained within the gap.

Throughout this paper, we assume a radial poloidal magnetic field, ${\rm{\Psi }}={\rm{\Psi }}(\theta )\propto -\cos \theta$ . The magnitude of Ψ is adjusted so that B = B_eq (Equation (2)) may be satisfied at $r=2{r}_{{\rm{g}}}$ . Assuming that the poloidal magnetic components dominate the toroidal one, we put $B{(r)={B}_{\mathrm{eq}}(r/2{r}_{{\rm{g}}})}^{-2}$ and compute the synchrotron emission outside the gap. Unless explicitly mentioned, we adopt $a=0.90{r}_{{\rm{g}}}$ , Ω_F = 0.50ω_H, ${R}_{\min }=6{r}_{{\rm{g}}}$ , j = 0.7, and R_c = r_g. To solve the Poisson Equation (19), we set the meridional boundary at θ = θ_max = 60°. We assume that the entire system is axisymmetric with respect to the rotation axis, which coincides with the magnetic axis.

5.1. The Case of Stellar-mass BHs

Let us first examine the case of M = 10 M_⊙. We begin with describing the 2D distribution of the ${E}_{\parallel }$ in Section 5.1.1, and demonstrate that the gap emission becomes strongest within the colatitude, θ < 38° along the magnetic axis in Section 5.1.2. Then in Sections 5.1.3–5.1.4, we examine the lepton densities and acceleration electric field. Adopting a distance of 1 kpc, we present the expected spectra of gap emissions in Section 5.1.5, and examine their dependence on j, a_*, and Ω_F in Sections 5.1.7–5.1.9. Finally, in Section 5.1.10, we demonstrate that the magnetosphere becomes entirely force-free (except for the gap region) by the cascaded pairs outside the gap.

5.1.1. Electric Field along the Magnetic Field Lines

We first present the distribution of the magnetic-field-aligned electric field on the poloidal plane. In Figure 3 we plot E_∥ (in statvolt cm⁻¹) as a function of the dimensionless tortoise coordinate, η_*, and the magnetic colatitude, θ (in degrees), for $\dot{m}=1.00\times {10}^{-4}$ .

$-\infty $ — **Figure 3.** Acceleration electric field (statvolt cm⁻¹) on the poloidal plane. The abscissa denotes the magnetic colatitudes, θ, in degrees, where 0 (i.e., the ordinate) corresponds to the magnetic axis. The ordinate denotes the dimensionless tortoise coordinate, where $-\infty$ corresponds to the event horizon. The gap is solved for a black hole with M = 10 M_⊙ and a_* = 0.9. The dimensionless accretion rate is chosen to be $\dot{m}$ = 1.00 × 10⁻⁴.
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We also plot ${E}_{\parallel }$ at six discrete colatitudes in Figure 4. It follows that the ${E}_{\parallel }({\eta }_{* })$ distribution changes little in the polar region within θ < 38°.

**Figure 4.** Acceleration electric field at six discrete colatitudes, θ, as a function of the dimensionless tortoise coordinate, ${\eta }_{* }$ , for $\dot{m}=1.00\times {10}^{-4}$ . The values of ${E}_{\parallel }({\eta }_{* },\theta )$ in Figure 3 are plotted at six discrete θs: the black solid, green dashed, blue dashed, magenta dotted, red dashed–dotted, and purple dashed–dotted curves denote the ${E}_{\parallel }$ at θ = 0°, 15°, 30°, 37.5°, 45°, and 52.5°, respectively.
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5.1.2. Gap Emission versus Colatitudes

We next compare the γ-ray spectra of a BH gap emission as a function of the colatitude, θ. In Figure 5, we compare the SEDs at the same six discrete θs as in Figure 4. It follows that the gap emission becomes most luminous if we observe the gap with a viewing angle θ < 38°. This conclusion is unchanged if we adopt different BH masses or spins. In what follows, we therefore adopt θ = 0 as the representative colatitude to estimate the maximum γ-ray flux of BH gaps.

**Figure 5.** Spectral energy distribution (SED) of the emission from a black hole (BH) gap at six discrete colatitudes, θ, at a distance of 1 kpc. The gap is solved for a black hole with M = 10 M_⊙ and a_* = 0.9. The dimensionless accretion rate is fixed at $\dot{m}=1.00\times {10}^{-4}$ . The six lines correspond to the same θs as in Figure 4.
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5.1.3. Created Lepton Densities

We plot the solved lepton densities at five discrete $\dot{m}$ in Figure 6. Since ${E}_{\parallel }$ is negative, electrons are accelerated outwards while positrons inwards. Thus, the dimensionless electronic density (solid curve), ${n}_{-}$ , per magnetic flux tube, increases outwards, while the positronic one (dashed curve), ${n}_{+}$ , decreases outwards. Note that the abscissa, $r-{r}_{0}$ , is converted from the tortoise coordinate to the Boyer–Lindquist radial coordinate for presentation purpose. Thus, $r-{r}_{0}=0$ corresponds to the null-charge surface. These solved ${n}_{\pm }(r,\theta )$ are used to compute the real charge density $\rho =({{\rm{\Omega }}}_{{\rm{F}}}B/2\pi c)({n}_{+}-{n}_{-})$ at each position, which is necessary to solve ${E}_{\parallel }$ on the poloidal plane. We continue iterations until ${n}_{\pm }({\eta }_{* },\theta )$ , ${E}_{\parallel }({\eta }_{* },\theta )$ , and the photon specific intensity saturate.

**Figure 6.** Spatial distribution of the electronic (solid) and positronic (dashed) densities per magnetic flux tube as a function of the distance, $r-{r}_{0}$ , from the null-charge surface, in the Boyer–Lindquist radial coordinate (in ${r}_{{\rm{g}}}\equiv {{GMc}}^{-2}$ unit). The gap is solved for a black hole with M = 10 M_⊙ and a_* = 0.9. Viewing angle is chosen to be θ = 0. The cyan, blue, green, black, and red curves represent the solutions for $\dot{m}=1.00\times {10}^{-3}$ , 3.16 × 10⁻⁴, 1.00 × 10⁻⁴, 7.49 × 10⁻⁵, and 5.62 × 10⁻⁵, respectively. The vertical dashed line at $r-{r}_{0}=0$ shows the null surface position.
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5.1.4. Acceleration Electric Field

As $\dot{m}$ decreases, the reduced ADAF near-IR photon field leads to less efficient pair production, thereby resulting in an extended gap to sustain the externally required current density, j, per magnetic flux tube. We plot ${E}_{\parallel }(r,\theta =0)$ for five discrete $\dot{m}$ s in Figure 7. The cyan, blue, green, black, and red curves correspond to the cases of $\dot{m}={10}^{-3.0}$ , ${10}^{-3.5}$ , 10^−4.0, 10^−4.125, and ${10}^{-4.25}$ , respectively; that is, the same as Figure 6. Integrating ${E}_{\parallel }$ over the gap width, we obtain the potential drop at each $\dot{m}$ . It becomes −6.3 × 10¹¹ V, −3.9 × 10¹² V, −2.5 × 10¹³ V, −4.2 × 10¹³ V, and −6.3 × 10¹³ V, for $\dot{m}={10}^{-3.0}$ , 10^−3.5, 10^−4.0, 10^−4.125, and 10^−4.25, respectively. Thus, the potential drop increases with decreasing $\dot{m}$ because of the increased gap width, $w\equiv {r}_{2}-{r}_{1}$ . More specifically, as the accretion rate reduces, the decreased ADAF near-IR photon field results in a less effective pair production for the gap-emitted IC photons, thereby increasing the mean-free path for two-photon collisions. Since w essentially becomes the pair-production mean-free path divided by the number of photons emitted by a single electron above the pair production threshold energy (Hirotani & Okamoto 1998), the reduced pair production leads to an extended gap along the magnetic field lines. As a result, the smaller $\dot{m}$ is, the greater the potential drop becomes.

**Figure 7.** Spatial distribution of the magnetic-field-aligned electric field, ${E}_{\parallel }$ , for five discrete accretion rates. The gap is solved for a black hole with M = 10 M_⊙ and a_* = 0.9. Each color corresponds to the same cases of $\dot{m}$ as in Figure 6. The vertical dashed line shows the null surface position.
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As w increases, the trans-field derivative begins to contribute in the Poisson Equation (19). As a result, the ${E}_{\parallel }$ distribution shifts outwards, in the same way as in pulsar outer-magnetospheric gaps (Figure 12 of Hirotani & Shibata 1999a). That is, a pulsar outer gap extends from the null surface to (or beyond) the light cylinder because the transverse thickness is limited by the efficient screening of ${E}_{\parallel }$ due to the trans-field propagation in concave poloidal magnetic field lines, whereas a BH gap shifts from the null surface toward the outer light surface because the longitudinal width becomes comparable to the transverse thickness when the accretion rate is small.

Let us briefly examine how the gap width, w, is affected when the ADAF soft photon field changes. In Figure 8, we plot the gap inner and outer boundary positions as a function of $\dot{m}$ , where the ordinate is converted into the Boyer–Lindquist radial coordinate. It follows that the gap inner boundary (solid curve, $r={r}_{1}$ ) infinitesimally approaches the horizon (dashed–dotted horizontal line, r = r_H), while the outer boundary (dashed curve, r = r₂) moves outwards, with decreasing $\dot{m}$ . Below the accretion rate $\dot{m}\sim {\dot{m}}_{\mathrm{low}}=2\times {10}^{-4}$ , the outer boundary moves rapidly away from the horizon with decreasing $\dot{m}$ , so that the required current density, j = 0.7, may be produced within the gap under a diminished ADAF photon field. At $\dot{m}=5.62\times {10}^{-5}$ , and 4.21 × 10⁻⁵, the outer boundary is located at ${r}_{2}=9.35{r}_{{\rm{g}}}$ and 28.64r_g. We consider that the solution with r₂ > 10r_g may not have significant physical meaning, because the funnel boundary with the equatorial disk will deviate from a conical shape beyond this radius (Hirose et al. 2004; McKinney & Gammie 2004; Krolik et al. 2005; McKinney 2006; McKinney et al. 2012; O' Riordan et al. 2016). We thus define the lower-limit accretion rate, ${\dot{m}}_{\mathrm{low}}$ when r₂ exceeds $10{r}_{{\rm{g}}}$ in this paper. Indeed, at further lower accretion rate, $\dot{m}\leqslant 3.16\times {10}^{-5}$ , we fail to find a 2D gap solution, because the weak ADAF photon field can no longer sustain the current density, j = 0.7, per magnetic flux tube. Thus, we obtain ${\dot{m}}_{\mathrm{low}}=5.56\times {10}^{-5}$ for M = 10 M_⊙, a = 0.90r_g, Ω_F = 0.50ω_H, and j = 0.7, interpolating ${r}_{2}={r}_{2}(\dot{m})$ and putting ${r}_{2}=10{r}_{{\rm{g}}}$ . Because r₂ rapidly increases near $\dot{m}\sim {\dot{m}}_{\mathrm{low}}$ , the value of ${\dot{m}}_{\mathrm{low}}$ depends little on whether we define it by e.g., r₂ = 10r_g or r₂ = 20r_g.

**Figure 8.** Spatial extent of the gap along the radial poloidal magnetic field line as a function of the dimensionless accretion rate. The gap is solved for a black hole with $M=10\,{M}_{\odot }$ and a_* = 0.9. Thick solid and dashed curves denote the position of the inner and the outer boundaries of the gap, in the unit of the gravitational radius, ${r}_{{\rm{g}}}\equiv {{GMc}}^{-2}$ . These boundary positions (in the ordinate) are transformed from the tortoise coordinate into the Boyer–Lindquist coordinate. The horizontal dashed–dotted line shows the horizon radius, whereas the horizontal dotted line shows the null surface position on the magnetic axis, θ = 0.
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5.1.5. Spectrum of Gap Emission

The predicted photon spectra are depicted in Figure 9 for the same set of $\dot{m}$ as in Figures 6 and 7. The thin curves on the left denote the input ADAF spectra, while the thick lines on the right show the output spectra from the gap. We find that the emitted flux increases with decreasing $\dot{m}$ , because the potential drop in the gap increases with decreasing $\dot{m}$ . The spectral peak around GeV is due to the curvature emission, while that around TeV is due to the IC scatterings. Provided that the distance is within several kpc, these HE and VHE fluxes appear above the Fermi/LAT detection limits (three thin solid curves labeled with "LAT 10 yrs"),⁴ and the CTA detection limits (dashed and dotted curves labeled with "CTA 50 hrs").⁵ Since BH transients spend the bulk of their time in a (very weakly accreting) quiescent state (e.g., Miller-Jones et al. 2011; Plotkin et al. 2013), it is possible for nearby BH transients to exhibit detectable BH emissions. For example, in HE, the flaring photons will be detectable if the duty cycle of the flaring activities is not too small (e.g., >0.1).

We plot the individual emission components in Figure 10, picking up the case of $\dot{m}=7.49\times {10}^{-5}$ (i.e., the case of the black solid line in Figure 9). The red dashed line shows the primary curvature component, and the red dashed–dotted line the primary IC component. The former component is not absorbed and appears as the spectral peak at several GeV when $\dot{m}\sim {\dot{m}}_{\mathrm{low}}$ (i.e., when the gap outer boundary is located at ${r}_{2}\gg {r}_{{\rm{g}}}$ ). The latter component is heavily absorbed by the ADAF near-IR photons to be reprocessed as the secondary component (blue dashed–dotted–dotted–dotted line). In this secondary component, IC emission dominates above GeV and the synchrotron component dominates only below this energy. The secondary component above 500 GeV is absorbed again to be reprocessed as the tertiary component (purple dotted).

To grasp the strength of the absorption taking place inside and outside the gap, we examine the optical depth for photon–photon collisions. For presentation purposes, we compute a representative optical depth for test photons emitted outwards at the gap inner boundary, $r={r}_{1}$ . With vanishing angular momenta, photons propagate in the (r, θ) surface (with constant θ). Thus, the invariant distance, ds, of a photon path after propagating dr and $d\varphi$ in r and φ coordinates becomes

$\begin{eqnarray}&&{{ds}}^{2}={g}_{{rr}}{{dr}}^{2}+{g}_{\theta \theta }d{\theta }^{2}=\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}{{dr}}^{2}\left[1+O\left(\displaystyle \frac{{\rm{\Delta }}{\sin }^{2}\theta }{{r}_{{\rm{g}}}^{2}}\right)\right].\end{eqnarray} \tag{ 27 }$

Thus, near the pole, $| \theta | \ll 1$ , and near the horizon, ${\rm{\Delta }}\ll {r}_{{\rm{g}}}^{2}$ , we can put ${ds}\approx \sqrt{{\rm{\Sigma }}/{\rm{\Delta }}}{dr}$ . Integrating the local absorption probability over the photon ray, we can compute the optical depth by

$\begin{eqnarray}&&\tau (\nu )={\int }_{{s}_{1}}^{\infty }\displaystyle \frac{d\tau }{{ds}}{ds}={\int }_{{r}_{1}}^{\infty }\sqrt{\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}}\displaystyle \frac{d\tau }{{ds}}{dr},\end{eqnarray} \tag{ 28 }$

where ds coincides with the distance of radial interval, dr, measured by ZAMO. The radial gradient of the optical depth becomes

$\begin{eqnarray}&&\displaystyle \frac{d\tau }{{ds}}=\displaystyle \frac{1}{c}{\int }_{0}^{1}d\mu ^{\prime} {\int }_{{\nu }_{\mathrm{th}}^{\prime }}^{\infty }\displaystyle \frac{d{\sigma }_{\gamma \gamma }(\nu ^{\prime} ,{\nu }_{{\rm{s}}}^{\prime },\mu ^{\prime} )}{d\mu ^{\prime} }\displaystyle \frac{{{dF}}_{{\rm{s}}}^{\prime }}{d{\nu }_{{\rm{s}}}^{\prime }}d{\nu }_{{\rm{s}}}^{\prime },\end{eqnarray} \tag{ 29 }$

where the primes denote quantities evaluated by ZAMO; $\mu ^{\prime}$ is the cosine of the collision angle of two photons, ${\nu }_{{\rm{s}}}^{\prime }$ the soft photon energy, and ${F}_{{\rm{s}}}^{\prime }$ the soft photon number flux. We employ ZAMO here, because the photons are assumed to be emitted with vanishing angular momenta in this paper. The threshold energy is defined by ${\nu }_{\mathrm{th}}^{\prime }$ = [2/(1 − μ')](m_ec)²/h²ν', where h is the Planck constant. The photon frequency ν (at infinity) is de-redshifted to the ZAMO value, ν', at each altitude, r. Specifically, the local photon energy, $h\nu ^{\prime}$ , is related to $h\nu$ , by $h\nu ^{\prime} =\dot{t}(h\nu +m\cdot d\varphi /{dt})$ , where m denotes the photon angular momentum and $d\varphi /{dt}$ the local observer's angular frequency in dt-basis at r. However, we here have m = 0, which gives the ZAMO angular frequency, $d\varphi /{dt}=-{g}_{t\varphi }/{g}_{\varphi \varphi }$ . The quantity $\dot{t}$ is given by the definition of the proper time, ${\dot{t}}^{2}[{g}_{{tt}}+2{g}_{t\varphi }d\varphi /{dt}+{g}_{\varphi \varphi }{(d\varphi /{dt})}^{2}]=-1$ , which reduces to $\dot{t}=\sqrt{{g}_{\varphi \varphi }}/{\rho }_{{\rm{w}}}$ for a ZAMO. Thus, we obtain $\nu ^{\prime} =\nu \sqrt{{g}_{\varphi \varphi }}/{\rho }_{{\rm{w}}}$ as the redshift relation between us and ZAMO. Note that the integral ${\int }_{{r}_{{\rm{H}}}}^{r}{{\rm{\Delta }}}^{-1/2}{dr}$ in Equation (28) is finite for a finite r, and that ${{dF}}_{{\rm{s}}}^{\prime} /d{\nu }_{{\rm{s}}}^{\prime }$ , and hence $d\tau /{ds}$ vanish at large distances.

In Figure 11, we present $\tau (\nu )$ for five discrete accretion rates: the five lines correspond to the same cases of $\dot{m}$ as in Figures 6, 7, and 9. It follows that the photon–photon absorption optical depth exceeds unity above 16 GeV, 90 GeV, 0.3 TeV, 0.5 TeV, and 0.9 TeV for $\dot{m}=1.00\times {10}^{-3}$ , 3.16 × 10⁻⁴, 1.00 × 10⁻⁴, 7.49 × 10⁻⁵, and 5.62 × 10⁻⁵, respectively. The optical depth peaks at several TeV, because the ADAF photon spectrum peaks in near-IR wavelengths for stellar-mass BHs.

$h\nu $ — **Figure 11.** Optical depth, τ, for photon–photon absorption as a function of the photon energy, $h\nu$ , at infinity for five $\dot{m}$ . A test photon is supposed to be emitted outwards with zero angular momentum from the gap inner boundary to infinity. The photon energy $h\nu$ is de-redshifted at each altitude, r, and the absorption probability gradient $d\tau /{ds}$ is integrated from the inner boundary to 60r_g, which is large enough.
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It should be stressed that the actual photons are absorbed by smaller optical depths than this figure, because the individual photons are emitted at different positions whose altitudes are always higher than the inner boundary.

5.1.6. Curvature versus Inverse-Compton Processes

It is worth examining the relative importance of the curvature and IC processes. In Figure 12, we plot the luminosity of the outward curvature-emitted photons as the solid curve, and that of the IC-emitted photons as the dashed–dotted one, as a function of $\dot{m}$ . The outward curvature photons are mostly emitted inside the gap, while the IC photons are emitted both inside and outside the gap. Figure 12 shows that the curvature luminosity exceeds the IC one when $\dot{m}\lt 2\times {10}^{-4}$ , or equivalently when the gap extends enough. This is because the curvature power is proportional to γ⁴, and the IC power approximately to γ⁰ ∼ γ², depending on whether the collisions take place mainly in the extreme Klein–Nishina or the Thomson regime. The IC power also depends on the specific intensity of the soft photon field; thus, its dependence on γ is more complicated than the curvature process.

5.1.7. Luminosity versus Created Current Density

Let us briefly examine the dependence of the spectrum on the created current density, j. In Figure 13, we plot the gap spectra at six discrete js. It follows that the HE emission becomes most luminous when 0.5 ≤ j ≤ 0.7 and the VHE one does when j ∼ 0.9. In what follows, we thus adopt j = 0.7 as a compromise to optimize the HE and VHE fluxes. Note that we restrict our argument for $| j| \leqslant 1$ , because $| j| \gt 1$ would incur a sign reversal of the $\rho -{\rho }_{\mathrm{GJ}}$ in Equation (19), thereby resulting in a sign reversal of ${E}_{\parallel }$ at the outer boundary, r = r₂, which would violate the present assumption of stationarity.

5.1.8. Luminosity versus BH Spin

We next briefly investigate how the spectrum depends on the BH spin parameter, ${a}_{* }=a/{r}_{{\rm{g}}}$ . In Figures 14 and 15, we present the gap spectra for ${a}_{* }=0.5$ and 0.998 when $\dot{m}=1.00\times {10}^{-4}$ . It follows that the gap flux increases with increasing BH spin. Thus, we adopt a_* = 0.9 as the representative value in this paper.

**Figure 14.** Similar to Figure 9, but the BH spin is chosen to be a_* = 0.5, while other parameters are the same, including the accretion rate, $\dot{m}$ .
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**Figure 15.** Similar to Figures 9 and 14, but the BH is maximally rotating, . Other parameters are the same as Figures 9 and 14, including the accretion rate.
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5.1.9. Luminosity versus Magnetic Field Rotation

It is numerically suggested that the angular frequency, Ω_F, of an accreting BH magnetosphere decreases from 0.3ω_H in the middle latitudes to −0.17ω_H in the higher latitudes (i.e., near the pole; McKinney et al. 2012). Similar tendency, from 0.4ω_H to −0.2ω_H is, indeed, analytically suggested (Beskin & Zheltoukhov 2013). Thus, in this subsection, we examine a smaller ${{\rm{\Omega }}}_{{\rm{F}}}$ case. In Figure 16, we plot the SED for Ω_F = 0.2ω_H, keeping other parameters unchanged from Figure 9. It follows that the HE flux decreases to about 17% of the Ω_F = 0.5ω_H case, while the VHE flux is roughly unchanged. The VHE flux changes mildly, because the IC process depends weakly on the lepton Lorentz factor compared to the curvature process. Note that the BZ power, which is proportional to Ω_F(ω_H − Ω_F), reduces to only 64% of the ${{\rm{\Omega }}}_{{\rm{F}}}=0.5{\omega }_{{\rm{H}}}$ case. This means that the gap becomes less efficient at smaller (in fact, also at greater) Ω_F/ω_H than 0.5. Thus, we adopt ${{\rm{\Omega }}}_{{\rm{F}}}=0.5{\omega }_{{\rm{H}}}$ as the representative value.

**Figure 16.** Similar to Figure 9, but the magnetosphere is more slowly rotating, Ω_F = 0.20ω_H. Other parameters are the same, including the accretion rate, $\dot{m}$ .
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5.1.10. Cascaded Pairs Outside the Gap

Let us investigate if a force-free magnetosphere is realized outside the gap. We compute the densities of the cascaded pairs within the cutting radius ${r}_{\mathrm{cut}}=60{r}_{{\rm{g}}}$ , which is well above the gap outer boundary, r = r₂. In Figure 17, we plot the secondary, tertiary, and quaternary pair densities as the red dashed, green dashed–dotted, and blue dashed–dotted–dotted–dotted curves, respectively. Here, the secondary pairs denote those cascaded from the primary γ-rays, which are defined as being emitted by the primary electrons or positrons that are accelerated in the gap. The tertiary pairs denote those cascaded from the secondary γ-rays, which are defined as being emitted by these secondary pairs.

**Figure 17.** Density of the cascaded pairs outside the gap, within radius r₂ < r < 60r_g, plotted as a function of $\dot{m}$ for M = 10 M_⊙ and a_* = 0.9. The red dashed, green dashed–dotted, and blue dashed–dotted–dotted–dotted curves denote the the densities of the secondary, tertiary, and quaternary, pairs, respectively. The black dotted curve shows the Goldreich–Julian charge density, which is volume-averaged within the sphere of $r=60{r}_{{\rm{g}}}$ .
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It follows that the density of the cascaded pairs exceeds the GJ one (black dotted line), as long as a stationary gap is formed. Thus, the magnetosphere becomes force-free outside the gap, as long as $\dot{m}\gt {\dot{m}}_{\mathrm{low}}$ . This conclusion does not depend on the choice of r_cut.

In short, gap solutions exist if the dimensionless accretion rate is in the range ${\dot{m}}_{\mathrm{low}}\lt \dot{m}\lt {\dot{m}}_{\mathrm{up}}$ . For stellar-mass BHs, the gap emission peaks at several GeV and its maximum flux >10⁻¹² erg s⁻¹ is detectable with Fermi/LAT, if the duty cycle of the flaring activity is not too small. The gap luminosity increases with decreasing $\dot{m}$ , because the gap is dissipating a portion of the BH's spin-down luminosity. This forms a striking contrast to accretion-powered systems, whose luminosity will decrease with decreasing $\dot{m}$ . The cascaded pairs outside the gap have a greater density than the Goldreich–Julian value; thus, the magnetosphere becomes force-free in the downstream of the gap-generated flow (i.e., outside the gap outer boundary).

5.2. The Case of Intermediate-Mass BHs

Let us examine the gap emission from intermediate-mass BHs. We put M = 10³ M_⊙ and calculate the gap emission for ${a}_{* }=0.9$ and Ω_F = 0.5ω_H. The predicted spectra become as in the top panel in Figure 18 for a distance of 10 kpc. It shows that the γ-ray fluxes are detectable in both HE and VHE if an IMBH is located within our galaxy, as long as the accretion rate is in the range, $2\times {10}^{-5}\lt \dot{m}\lt 4\times {10}^{-5}$ .

Since the absolute luminosity of the gap increases with M, we also compute the SEDs for M = 10⁵ M_⊙ (bottom panel of Figure 18), assuming d = 100 kpc. It shows that such a heavy BH is detectable in HE or VHE only when the distance is comparable to or less than 100 kpc.

The curvature and IC luminosities are plotted as a function of $\dot{m}$ in Figure 19. It shows that the curvature process is stronger than the IC when $\dot{m}\lt 6\times {10}^{-5}$ for M = 10³ M_⊙, and when $\dot{m}\lt 1.5\times {10}^{-5}$ for M = 10⁵ M_⊙.

**Figure 19.** Top panel: gap luminosity as a function of the dimensionless accretion rate for a BH with $M={10}^{3}\,{M}_{\odot }$ , a_* = 0.9, and Ω_F = 0.5ω_H. The solid and dashed–dotted curves denote the luminosity of the curvature and inverse-Compton processes, respectively. Bottom panel: Similar figure to the top panel but for M = 10⁵ M_⊙.
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We examine if a force-free magnetosphere is sustained. Figure 20 shows the densities of the pairs cascaded between r₂ and $60{r}_{{\rm{g}}}$ . It follows that the density of the cascaded pairs exceeds the GJ density for both M = 10³ M_⊙ and M = 10⁵ M_⊙.

5.3. The Case of SMBHs

Next, let us examine SMBHs. We present the acceleration electric field in Section 5.3.1, gap spectrum in Section 5.3.2, and cascaded pair densities in Section 5.3.3. We briefly examine the dependence on the soft photon density in Section 5.3.4.

5.3.1. Acceleration Electric Field

Choosing a typical mass of M = 10⁹ M_⊙, we plot an ${E}_{\parallel }({\eta }_{* },\theta )$ distribution in Figure 21, whose ordinate is the dimensionless tortoise coordinate, ${\eta }_{* }$ . Comparing this with Figure 3, we find that the essential behavior of ${E}_{\parallel }$ is unchanged from the case of M = 10 M_⊙. In the Boyer–Lindquist radial coordinate, the gap outer and inner boundaries are distributed as a function of $\dot{m}$ as depicted in Figure 22. The gap outer boundary is located at ${r}_{2}=5.60{r}_{{\rm{g}}}$ and r₂ = 10.32r_g at $\dot{m}=7.49\times {10}^{-7}$ and 5.62 × 10⁻⁷, respectively; thus, we obtain ${\dot{m}}_{\mathrm{low}}=5.75\times {10}^{-7}$ for M = 10⁹ M_⊙, a = 0.90r_g, ${{\rm{\Omega }}}_{{\rm{F}}}=0.50{\omega }_{{\rm{H}}}$ , and j = 0.7. Figure 22 shows that the gap longitudinal width becomes comparable or greater thanr_g when the accretion rate reduces to $\dot{m}\lt 2\times {10}^{-6}$ , whereas it is realized when $\dot{m}\lt 2\times {10}^{-4}$ for a stellar-mass case (Figure 8). It also follows that the gap outer boundary shifts outwards with decreasing $\dot{m}$ , in the same manner as the M = 10 M_⊙ case.

**Figure 21.** Similar to Figure 4, but for M = 10⁹ M_⊙. The dimensionless accretion rate is chosen to be $\dot{m}=1.00\times {10}^{-6}$ .
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**Figure 22.** Similar to Figure 8, but for a supermassive black hole with M = 10⁹ M_⊙ and ${a}_{* }=0.9$ .
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**Figure 22.** Similar to Figure 8, but for a supermassive black hole with M = 10⁹ M_⊙ and ${a}_{* }=0.9$ .
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5.3.2. Spectrum of Gap Emission

Figure 23 shows the gap spectra as the thick lines for five discrete $\dot{m}$ , assuming a luminosity distance of 10 Mpc. When the accretion rate is in the range $5.6\times {10}^{-7}\lt \dot{m}\lt {10}^{-6}$ , we find that the gap emission will be marginally detectable with CTA, if the source is located in the southern sky. The emission components are depicted in Figure 24 for $\dot{m}=7.49\times {10}^{-7}$ . Since $\dot{m}$ is much smaller than the stellar-mass cases, the absorption optical depth decreases accordingly; as a result, most of the primary IC component (red dashed–dotted line) cascades only to the secondary generation pairs, whose emission is represented by the blue dashed–dotted–dotted–dotted line.

**Figure 23.** SED of the gap for a supermassive BH with M = 10⁹ M_⊙ and a_* = 0.9 at 10 Mpc. The cyan, blue, green, black, and red curves represent the SEDs for $\dot{m}=1.00\times {10}^{-5}$ , 3.16 × 10⁻⁶, 1.00 × 10⁻⁶, 7.49 × 10⁻⁷, and 5.62 × 10⁻⁷, respectively.
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**Figure 24.** Similar to Figure 23, but only the case of $\dot{m}=7.49\times {10}^{-7}$ is depicted. The thick and thin black curves correspond to the same ones in Figure 23. The red dashed and dashed–dotted lines denote the primary curvature and inverse-Compton components, respectively. The blue dashed–dotted–dotted–dotted line is secondary emission via synchrotron and inverse-Compton processes outside the gap.
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**Figure 24.** Similar to Figure 23, but only the case of $\dot{m}=7.49\times {10}^{-7}$ is depicted. The thick and thin black curves correspond to the same ones in Figure 23. The red dashed and dashed–dotted lines denote the primary curvature and inverse-Compton components, respectively. The blue dashed–dotted–dotted–dotted line is secondary emission via synchrotron and inverse-Compton processes outside the gap.
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The curvature and IC luminosities are plotted as a function of $\dot{m}$ in Figure 25. It is clear that the IC process dominates the curvature one in the entire range of $\dot{m}$ for SMBHs.

5.3.3. Cascaded Pairs Outside the Gap

The created pair densities between r₂ and 60r_g are depicted as a function of $\dot{m}$ in Figure 26. In the same way as for stellar-mass BHs (Section 5.1) and intermediate-mass BHs (Section 5.2), the magnetosphere becomes force-free outside the gap, as long as the gap solution exists.

Figures 17, 20 and 26 show that the BH magnetosphere becomes force-free for M = 10 M_⊙, 10³ M_⊙, 10⁵ M_⊙, and 10⁹ M_⊙. Indeed, a BH magnetosphere becomes force-free irrespective of the BH mass, as long as a stationary gap is sustained in it by $\dot{m}\gt {\dot{m}}_{\mathrm{low}}(M)$ .

5.3.4. Dependence on RIAF Soft Photon Field Density

Let us quickly examine how the gap luminosity depends on the ADAF photon field density, fixing $\dot{m}$ so that the magnetic field near the horizon may not be changed. As a test model, we artificially reduce the photon density near the horizon to one fourth of its original value by doubling R_min. In Figure 27, we plot the gap spectra for R_min = 12r_g. We find that the gap emission increases because of the diminished soft photon density, as expected. To predict the gap spectrum of SMBHs more precisely, we must constrain the specific intensity of the RIAF photon field near the horizon for example by numerical simulations.

**Figure 27.** Similar to Figure 23, but the ADAF photon field is artificially reduced to one fourth by adopting ${R}_{\min }=12{r}_{{\rm{g}}}$ , instead of R_min = 6r_g.
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**Figure 27.** Similar to Figure 23, but the ADAF photon field is artificially reduced to one fourth by adopting ${R}_{\min }=12{r}_{{\rm{g}}}$ , instead of R_min = 6r_g.
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5.4. Gap Luminosity for Various BH Masses

Finally, we apply the method to various BH masses from 10 M_⊙ to 6.4 × 10⁹ M_⊙. The filled circles in Figure 1 denote the solved ${\dot{m}}_{\mathrm{low}}(M)$ for $({a}_{* },{{\rm{\Omega }}}_{{\rm{F}}}/{\omega }_{{\rm{H}}},{R}_{\min }/{r}_{{\rm{g}}})=(0.9,0.5,6)$ . It follows that ${\dot{m}}_{\mathrm{low}}(M)$ always lies below the upper limit, ${\dot{m}}_{\mathrm{up}}$ (upper straight line). Therefore, BH gaps exist for arbitrary BH masses.

We fit the filled circles in Figure 1 with a straight line by the least square method to obtain $\mathrm{lg}\,{\dot{m}}_{\mathrm{low}}=-3.6\mbox{--}0.26\mathrm{lg}(M/{M}_{\odot })$ . The luminosity, L_gap, of a BH gap, could be estimated by substituting this $\dot{m}\approx {\dot{m}}_{\mathrm{low}}$ into L_gap ≈ 0.3 L_BZ ≈ 1.5 × 10³⁸ $\dot{m}{M}_{1}\,{\text{erg s}}^{-1}$ , where Equation (3) is used. The factor 0.3 comes from the fact that the gap luminosity can attain at most ∼30% of L_BZ, because E_∥ is partly screened by the created pairs within the gap, and because a stationary gap solution can possess only a sub-GJ current density. It is also noteworthy that the BH gap solution at such a low $\dot{m}$ corresponds to the case of the middle-aged pulsars whose outer-gap luminosity attains at most 30% of the spin-down luminosity for the same reasons (Hirotani 2013).

If we artificially change R_min, we can consider the impact of the ADAF photon field density, without changing B at the same $\dot{m}$ . In Figure 1, we plot the ${\dot{m}}_{\mathrm{cr}}(M)$ for (a_*, Ω_F/ω_H, R_min/r_g) = (0.9, 0.5, 12) as the open squares. It follows that the reduction of the soft photon field (by doubling R_min) changes the gap solution to some extent, because it affects the pair production process in the gap.

In Figure 1 we also plot a slower BH spin, $({a}_{* },{{\rm{\Omega }}}_{{\rm{F}}}/{\omega }_{{\rm{H}}},{R}_{\min }/{r}_{{\rm{g}}})=(0.5,0.5,6)$ as the open circles. It follows that ${\dot{m}}_{\mathrm{low}}(M)$ has a weak dependence on a_* due to the negative feedback effect (Hirotani 2013). To further constrain ${\dot{m}}_{\mathrm{low}}(M)$ , we need to specify the specific intensity of the RIAF (e.g., ADAF) photon field near the horizon by numerical computation. Such details are, however, out of scope of the present paper. Therefore, the region $\dot{m}\lt \dot{m}(M)$ (in which stationary gaps do not exist) is depicted in red gradation in Figure 1.

6. DISCUSSION

To sum up, we have solved stationary lepton accelerators (or gaps) in the magnetospheres of rotating BHs with arbitrary masses. By solving the set of an inhomogeneous part of the Maxwell equations, lepton equations of motion, and the radiative transfer equation, we demonstrate that an electric field arises along the magnetic field line around the null charge surface on which the frame-dragging angular frequency coincides with the magnetic-field angular frequency. In the gap, electrons and positrons are created via two-photon collisions and accelerated in opposite directions by the acceleration electric field into ultra-relativistic energies. Such leptons emit copious γ-rays via curvature and inverse-Compton (IC) processes, leading to a pair-production cascade inside and outside the gap. The gap longitudinal width is self-regulated so that a single electron eventually cascades into a single pair within the gap, and approximately coincides with the mean-free path (for an IC photon to materialize via two-photon collision) divided by the number of IC photons emitted by a single electron. As the accretion rate decreases, the increased mean-free path results in an extended gap, and hence an increased luminosity. The gap luminosity maximizes when the gap width becomes much greater than the horizon radius. For stellar-mass BHs, we can expect that their curvature emission is detectable with Fermi/LAT, and the IC emission with CTA, when the BH binary is in a quiescent state, provided that the distance is within several kpc. For SMBHs, their IC emission may be marginally detectable with CTA for low-luminosity AGN if their distances are within a few tens of Mpc.

6.1. Improvement form HP16

In the present work, there are mainly two improvements from HP16, who formulated a BH gap model and applied it to the radio galaxy IC310. First, in the present work, the Poisson Equation (15) is solved general-relativistically on the poloidal plane. However, in HP16, the left-hand side of Equation (15) was approximated one-dimensionally in the Newtonian limit. Second, in the present work, $\rho (r,\theta )$ (in the Poisson equation) is solved from the local pair production rate in a manner consistent with the radiative transfer equation. However, HP16 assumed that $\rho (r)$ changes linearly with r (see their Equation (10)).

Both above-mentioned simplifications suppress E_∥ to some extent, when comparing the GR treatment. To compare the difference, we apply the present 2D, GR method to IC310, assuming B = B_eq. Figure 28 shows the resultant SED, where the parameter set is basically the same as HP16; namely, we adopt ${a}_{* }=0.998$ and Ω_F = 0.3ω_H. Compared with Figure 13 of HP16, which was obtained for $B={B}_{\mathrm{eq}}$ , we find that the 1D Newtonian approximation in HP16 underestimated the gap luminosity about 1.5 times from the present 2D GR treatment.

**Figure 28.** BH gap emission spectrum of IC310*. Extreme spin, ${a}_{* }=0.998$ , is assumed to compare with Figure 13 (B = B_eq case) of HP16. The cyan, blue, green, and black curves correspond to $\dot{m}=1.00\times {10}^{-5}$ , 3.16 × 10⁻⁶, 1.00 × 10⁻⁶, and 7.49 × 10⁻⁷, respectively. The black triangles and squares show the observed fluxes in the low and high states (Aleksić et al. 2010) respectively, while the red circles do those in the flaring state (Aleksić et al. 2014b). The observed VHE fluxes are corrected for the absorption by the extragalactic background light. The solid bowtie represents the X-ray fluxes obtained by re-analyzing the observational data taken with *XMM-Newton* in 2003, *Chandra* in 2004, and *Swift* in 2007 (Aleksić et al. 2014a). The dotted bowtie represents the re-analyzed data of *Chandra* in 2005 (Aleksić et al. 2014a). The open circles denote the *Fermi*/LAT spectrum.
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**Figure 28.** BH gap emission spectrum of IC310*. Extreme spin, ${a}_{* }=0.998$ , is assumed to compare with Figure 13 (B = B_eq case) of HP16. The cyan, blue, green, and black curves correspond to $\dot{m}=1.00\times {10}^{-5}$ , 3.16 × 10⁻⁶, 1.00 × 10⁻⁶, and 7.49 × 10⁻⁷, respectively. The black triangles and squares show the observed fluxes in the low and high states (Aleksić et al. 2010) respectively, while the red circles do those in the flaring state (Aleksić et al. 2014b). The observed VHE fluxes are corrected for the absorption by the extragalactic background light. The solid bowtie represents the X-ray fluxes obtained by re-analyzing the observational data taken with *XMM-Newton* in 2003, *Chandra* in 2004, and *Swift* in 2007 (Aleksić et al. 2014a). The dotted bowtie represents the re-analyzed data of *Chandra* in 2005 (Aleksić et al. 2014a). The open circles denote the *Fermi*/LAT spectrum.
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6.2. Rigidity of Rotating Magnetic Field Lines

We have assumed that ${{\rm{\Omega }}}_{{\rm{F}}}\equiv {F}_{{tr}}/{F}_{r\varphi }={F}_{t\theta }/{F}_{\theta \varphi }$ is constant along each magnetic field line. However, Ω_F is not conserved when the electric potential drop along the magnetic field line becomes a non-negligible fraction of the electro-motive force exerted across the horizon. Nevertheless, for all the cases in the present paper, the gap luminosity is less than two percent of the Blandford–Znajek power. Noting that the particle energy is predominantly converted into radiation within the gap, we can conclude that the deviation of Ω_F from a constant value (e.g., 0.5ω_H) is at most a few percent. Thus, the assumption of a constant Ω_F along each magnetic field line is mostly justified.

6.3. Gap Position versus the Separation Surface

It is worth comparing the gap position with the separation surface where both inflows and outflows start without E_∥. Since the plasma particles will not be accelerated either outward or inward at this surface, the sum of the gravitational, Lorentz and centrifugal forces will vanish along the magnetic field lines there. One of the convenient ways to find such a surface is to put both u_p = 0 and ${u}_{{\rm{p}}}^{\prime }=0$ in the poloidal wind equation of MHD (Camenzind 1986a, 1986b), where u_p refers to the poloidal velocity of the fluid. Note that this MHD argument is valid in the magnetically dominated limit, which is close to (but not equal to) the force-free limit.

In a stationary and axisymmetric BH magnetosphere, the separation surface is determined by the condition k₀' = 0 in MHD (see end of Section 3; Takahashi et al. 1990), where the prime denotes the derivative along the poloidal magnetic field line. For radial poloidal field lines, k₀ attains its maximum value 0.651, 0.478, 0.372, and 0.310 at r = 8.34r_g, 5.21r_g, 4.02r_g, and 3.45r_g, along the field lines at θ = 15°, 30°, 45°, and 60°, respectively. Thus, the separation surface is approximately located at 5r_g < r < 8r_g at θ < 30°.

As for the gap position, Figures 3, 4, and 21 show that ${E}_{\parallel }(r,\theta )$ distribution, and hence the gap longitudinal extent depends on little θ, except near the meridional boundary, $\theta \sim {\theta }_{\max }=60^\circ$ . Thus, near the magnetic pole (e.g., $| \theta | \lt 30^\circ$ ), the separation surface is generally located outside the gap (Figures 8 and 22). This conclusion depends little on the poloidal magnetic field structure, because the gap solution is essentially determined by the ${\rho }_{\mathrm{GJ}}$ distribution, which is governed by the radial component of the magnetic field near the horizon (Equation (24)) and because the separation surface distribution has a weak dependence on the poloidal field structure as well (Figure 2 of HP16).

As a plasma fluid begins to flow with u_p ≈ 0 from the separation surface, it is accelerated along the flow line by the MHD interactions; however, their Lorentz factors can attain only a few (e.g., Equation (5.2) of Hirotani et al. 1992). On the other hand, the electron–positron pairs that cascaded from outward-propagating PeV γ-rays have typical Lorentz factors ∼10⁶ (for stellar-mass BHs). Thus, such pairs continue their outward motion across the separation surface, climbing up the "hill" of the MHD effective potential k₀ (Figure 2 of HP16) very easily. In other words, the large charge-to-mass ratio of the electron makes the electrostatic acceleration dominate the MHD acceleration, which is comparable to the centrifugal or gravitational one. Only in the direct vicinity of the horizon, do the plasma mass and the causality at the horizon make the gravitational interaction overcome the electromagnetic one. Put differently, we can forget about the separation surface when we consider the formation of a gap in a BH magnetosphere.

It is worth noting that the above-mentioned MHD separation surface is distinct from what is argued in recent MHD simulations. In numerical simulations, the flow density decreases to zero at a specific surface, which requires an imposition of "density floors." Without the density floors, a vacuum would develop in the funnel. However, such a limitation of numerical MHD has nothing to do with the gap formation discussed in the present paper.

6.4. The Case of Very Small Accretion Rate

Let us discuss what is expected when $\dot{m}$ becomes even less than ${\dot{m}}_{\mathrm{low}}(M)$ . The ADAF photon field peaks around eV for stellar-mass BHs and around meV for SMBHs. These photons are emitted from the innermost region, r ∼ R_min ∼ 6r_g, and decreases outward approximately by r⁻² law. Thus, when the gap outer boundary is located at r ≫ R_min, as in the case of $\dot{m}\sim {\dot{m}}_{\mathrm{low}}(R)$ , stationary pair production can be sustained only marginally in an extended gap. However, at $\dot{m}\lt {\dot{m}}_{\mathrm{low}}(R)$ , stationary pair production can be no longer sustained and a vacuum region develops in the entire polar funnel. In this vacuum region, migratory leptons are accelerated by the vacuum ${E}_{\parallel }$ and cascade into copious primary electrons and positrons accelerated in opposite directions. Emissions from such cascading primaries will be time-dependent and may consist of many "shots," whose power spectrum density (PSD) may show some characteristic power law. In this case, the lower cutoff frequency of the power-law PSD will be given by the reciprocal of the light crossing time of the extended gap, which is much longer than the horizon-light-crossing timescale.

6.5. Distinction Between Gap and Jet Emissions

We finally discuss how to discriminate the gap and jet emissions. It follows from Figure 9 that the gap HE and VHE fluxes increase with decreasing $\dot{m}$ . That is, we can predict an anti-correlation between the IR/optical and HE/VHE fluxes. It forms a contrast to the standard shock-in-jet scenario, in which the IR/optical and the HE/VHE fluxes will correlate. Therefore, we propose to simultaneously observe nearby BH transients during quiescence both in near-IR/optical and VHE. If their time-varying multi-wavelength spectra show anti-correlation, it strongly suggests that the photons are emitted from the BH gap. For nearby low-luminosity AGNs, the anti-correlation will appear between submillimeter wavelength and VHE.

In X-rays, the gap emission is very weak. Thus, if X-ray photons are detected, they are probably emitted from the jet or from the accretion flow.

Stellar-mass BHs exhibit the strongest gap emission in HE. Figure 9 shows that nearby BH transients are capable of emitting an order of magnitude greater fluxes than the LAT detection limit (with 10 years observation). It means that an HE "flare" could be detected by LAT if the flare lasts for a month or so, particularly for the sources located away from the galactic plane. We will examine the plausible sources in a subsequent paper.

One of the authors (K. H.) is indebted to Dr. T. Y. Saito for valuable discussion on the CTA sensitivity, and to Drs. K. Kashyama, K. Asada, M. Nakamura, A. K. Harding, D. Kazanas, S. Shibata for fruitful discussion. This work is supported by the Theoretical Institute for Advanced Research in Astrophysics (TIARA) operated under Academia Sinica.