Multi-dimensional simulations of ergospheric pair discharges around black holes
Benjamin Crinquand, Benoît Cerutti, Alexander Philippov, Kyle Parfrey, Guillaume Dubus
MMulti-dimensional simulations of ergospheric pair discharges around black holes
Benjamin Crinquand, ∗ Benoît Cerutti, Alexander Philippov, Kyle Parfrey, and Guillaume Dubus Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA (Dated: March 10, 2020)Black holes are known to launch powerful relativistic jets and emit highly variable gamma radiation. Howthese jets are loaded with plasma remains poorly understood. Spark gaps are thought to drive particle acceler-ation and pair creation in the black-hole magnetosphere. In this paper, we perform 2D axisymmetric general-relativistic particle-in-cell simulations of a monopole black-hole magnetosphere with a realistic treatment ofinverse Compton scattering and pair production. We find that the magnetosphere can self-consistently fill itselfwith plasma and activate the Blandford-Znajek mechanism. A highly time-dependent spark gap opens nearthe inner light surface which injects pair plasma into the magnetosphere. These results may account for thehigh-energy activity observed in active galactic nuclei and explain the origin of plasma at the base of the jet.
Active galactic nuclei (AGN) can be responsible for thelaunching of powerful relativistic plasma jets. Very longbaseline interferometry shows that these jets are launchedvery close to the event horizon of the black hole [1], imply-ing that processes occurring in its close environment mustbe at play. Some AGN are also known to emit ultra-rapidgamma-ray flares [2, 3] suggesting that sub-horizon scales,possibly at the base of the jet, are involved in e ffi cient par-ticle acceleration. Non-thermal emission from acceleratedparticles was recently detected in the immediate vicinity ofthe AGN M87* [4]. This creates new opportunities to betterunderstand black-hole activity, as the black-hole system cannow be directly probed down to sub-horizon scales.A possible explanation for jet launching is provided bythe Blandford-Znajek (BZ) mechanism [5], which involvesa force-free magnetosphere coupled to the black hole. Thismechanism requires plasma to be continuously replenished,in order to sustain the force-free magnetosphere and to carrythe Poynting flux. The jet generally comprises the magneticfield lines which enter the ergosphere and cross the eventhorizon. Since these field lines are disconnected from thedisk, it is very unlikely that plasma from the accretion flowcan fill the jet zone.As the plasma density drops, the electric field induced bythe rotation of the black hole becomes unscreened, leadingto electrostatic gaps and particle acceleration. High-energyemission may result from inverse Compton (IC) scattering ofsoft photons by ultra-relativistic leptons. In this framework,annihilation between the high-energy photons produced inthe gap and soft photons emitted by the accretion flow isa possible plasma source [6]. Electrostatic gaps could thenboth explain the observed gamma-ray flares and provide pairplasma to the jet.There have been numerous attempts to derive analyticallythe properties of a steady gap [7, 8], but the spark gap dy- ∗ Correspondence email address: [email protected] namics are most likely intermittent [9]. The exact locationof the gap is also unknown. The validity of the BZ mecha-nism has been demonstrated by general relativistic magneto-hydrodynamic (GRMHD) simulations [e.g. 10], but this nu-merical approach cannot address the questions of the sourceof plasma or particle acceleration. Kinetic simulations, onthe other hand, can capture these e ff ects. 1D general rela-tivistic particle-in-cell (GRPIC) simulations display a time-dependent gap [11, 12]. Parfrey et al. [13, hereafter P19]performed the first global 2D GRPIC simulations of a nearlyforce-free magnetosphere. They ignored radiative transferand instead injected pairs in proportion to the local paral-lel electric field. This prescription mimics pair creation, butprecludes any chance of seeing a gap develop.In this work, we present 2D global GRPIC simulationswith self-consistent radiative transfer, in order to model re-alistic plasma injection and study the spark gap dynamics.Both IC scattering and γγ pair creation processes are imple-mented. We use a general relativistic version of the PIC code Zeltron [14–16], first introduced in P19. The backgroundspace-time is described by the Kerr metric, with dimension-less spin parameter a ∈ [0 , t , r , θ, ϕ ), which do not possess a coordinatesingularity at the event horizon. For convenience, we define“fiducial observers” (FIDOs), whose wordlines are orthogo-nal to spatial hypersurfaces.We include gamma-ray photons in our simulations as aneutral third species that follows null geodesics. We ex-tended to full 3D the radiative transfer algorithm of Levin-son and Cerutti [11], which incorporates IC scattering andphoton-photon pair production (see the Supplemental Ma-terial, which includes Refs. [17–22]). Electrons, positronsand gamma-ray photons interact with a background radia-tion field of soft photons. For simplicity, we assume that theradiation field is time-independent, uniform, isotropic andmono-energetic, with energy ε and density n . We do notinclude any feedback of the simulation on this radiation field.The upscattered photons and created leptons are assumed topropagate along the same direction as their high-energy par- a r X i v : . [ a s t r o - ph . H E ] M a r ents, reflecting strong relativistic beaming. The fiducial op-tical depth of both processes is τ = n σ T r g , where r g is thegravitational radius and σ T is the Thomson cross-section.In this paper we choose to endow the black hole witha monopole magnetic field (see the Supplemental Mate-rial). Although unphysical, this magnetic configuration hasseveral benefits. (i) Our results can be directly comparedto the BZ analytical solution, which assumes a magneticmonopole. (ii) We can capture the intrinsic physical prop-erties of the gap without interference from more complexstructures, such as current sheets. (iii) It is a realistic modelfor the field lines penetrating the ergosphere on each hemi-sphere, irrespective of the magnetosphere’s large-scale struc-ture [10, 23].We use a 2D axisymmetric setup with spherical coor-dinates ( r , θ ). The simulation domain is r ∈ [ r min = . r h , r max = r g ], θ ∈ [0 , π ], where r h = r g (1 + √ − a )is the radius of the event horizon. The ergosphere is the re-gion within the axisymmetric surface defined by r = r g (1 + √ − a cos θ ). The spin parameter is set at a = . r and θ . We mimic an open outer boundary using an absorbingboundary layer [16]. Particles are removed if r ≤ r h or r ≥ r max . We performed our runs with a grid resolution2000 ( r ) × θ ), with the requirement that we resolvethe plasma skin depth everywhere. This was checked a pos-teriori since the plasma density is one of the unknowns. Ini-tially, the magnetosphere is empty of pairs but filled withgamma-ray photons distributed uniformly and isotropicallyfrom r = r h to r = r h , with the energy ε = m e c ,which is well above the pair creation threshold. The photonsquickly pair produce, igniting the pair discharge.We use normalized code units where r g is the unit oflength and r g / c the unit of time. The normalized magneticfield is ˜ B = r g ( eB / m e c ), and the normalized energy ofbackground photons is ˜ ε = ε / m e c . Three dimensionlessparameters define the physical conditions around the blackhole: ˜ B , ˜ ε and τ . In M87*, the magnetic field is estimatedto be B ≈
100 G ( ˜ B ∼ ) [24, 25], whereas the softbackground photon field peaks at ε ≈ ε ∼ − )[26]. The optical depth is uncertain, but is likely to be (cid:46) [6, 27]. The density scale needed to screen the vacuum paral-lel electric field is the typical Goldreich-Julian number den-sity n GJ = B ω BH / (4 π ce ) [28], taking ω BH = ca / (2 r h ) as theblack-hole angular velocity.The maximum Lorentz factor γ max that leptons can reachis close to a ˜ B . We also define γ s as the typical Lorentzfactor of secondary particles that have just been pair pro-duced. We focus our work on AGN characterized by1 (cid:28) γ s (cid:28) γ max . The cross-section of γγ pair produc-tion peaks near the threshold [20], so the bulk of pairs iscreated at γ s ∼ / ˜ ε . The greater the ratio γ max /γ s ∼ a ˜ B ˜ ε ,the higher the resulting plasma multiplicity (defined as theplasma density normalized by n GJ ) will be [29].Altogether, we must choose ˜ ε low enough, so that γ s (cid:29)
1, but ˜ B ˜ ε large enough, to guarantee a good sep-aration of scales and a large multiplicity ( γ s (cid:28) γ max ). Inpractice we chose ˜ B = × and ˜ ε = × − . Theproduct ˜ B ˜ ε = B ˜ ε , the gaps remain steady at all τ . On the other hand,increasing the magnetic field implies decreasing the plasmaskin depth d e = (cid:112) m e c / π n GJ e ∼ r g ˜ B − / , so the resolu-tion needs to go up. We are thus limited to unrealisticallylow values for ˜ B and high values for ˜ ε , since ˜ B ˜ ε mustremain large.Our simulations have Ω · B > Ω · B < Ω is the black-hole an-gular velocity vector. In order to screen the electric field, theblack-hole magnetosphere requires a negative poloidal cur-rent in the upper hemisphere ( z >
0) and a positive currentin the lower hemisphere ( z < z >
0, and lower for z <
0. Still, the plasma remains globally neutral during thesimulation. A species in the upper hemisphere has the samebehavior as its anti-species in the lower hemisphere. Parti-cles flow mainly radially, along the magnetic field lines. Weran four simulations with τ =
5, 10, 20 and 30. A steadystate is reached after around 50 to 100 r g / c , as determinedby the total number of particles in the box.We observe a transition between two regimes with in-creasing τ (Fig. 1). At low optical depths ( τ (cid:46) τ (cid:38) ffi ciently. It is extremely intermit-tent, ejecting shreds of pair plasma outwards (see the insetof Fig 1 for τ = D · B / B , ranges between 10 − and10 − , which is similar to the ad hoc values used in P19. Inter-mediate opacity simulations display an intermediate regime:high latitude field lines behave similarly to the low opacitycase (see Fig. 1 for τ = n for positrons (left) and electrons (right),compensated by r , for three fiducial optical depths τ =
10, 20 and 30. Insets show the density close to the horizon.Bottom panel: snapshots of the steady-state radial 3-velocities v r for positrons (left) and electrons (right) for τ =
10, 20and 30. The loosely dashed black line is the stagnation surface given by [8]. Insets show the 3-velocity close to the blackhole, with higher contrast to help visualize the change in sign. The two solid lines are the inner and outer light surfaces. Thedotted black line is the null surface as given by [9]. In all plots, the densely dashed red line marks the ergosphere. Alldistances are in units of r g .the equator show the same time-dependent behavior as thehigh opacity run. The inner and outer light surfaces, be-yond which the rotation of magnetic field lines is superlumi-nal [30], are shown on the lower plots in Fig. 1. Their shapesat high opacity are consistent with what was previously de-rived in the force-free regime [e.g. 30, 31]. The size of thesimulation box was set so as to include both light surfaces.The insets in the lower panels of Fig. 1 show the radialcomponent of the electron 3-velocity near the horizon. Fo-cusing on the upper hemisphere only, in all simulations thereis an electron velocity separation surface located exactly atthe inner light surface. The positron velocity separation sur-face has a di ff erent location, which depends on the opacity.It always lies between the inner and outer light surfaces. Thehigher τ , the closer to the black hole the positron separationsurface is. The situation is symmetric (switching positronsand electrons) in the lower hemisphere. The high opacity simulations present similarities with the low plasma supplysimulation in P19, in particular regarding the role of the lightsurface. However, in our simulations all particles fly awayfrom the black hole outside of the outer light surface, as aresult of the di ff erent magnetic configuration used. Withinthe inner light surface, both species fall into the black holefor all τ . We ran a simulation at high opacity but with spin a = .
75 and confirmed that the inner light surface retainsthe same role.The MHD stagnation surface, separating inflow and out-flow in single-fluid MHD [32], has been suggested as a plau-sible position for the gap [7]. Its location can be derivedanalytically [8] and is presented in the top panel of Fig. 1.The null surface is where the general-relativistic Goldreich-Julian charge density vanishes [9] and has also been pro-posed as as plausible gap position. We find that both thestagnation surface and the null surface are irrelevant for thepair discharge, and that the inner light surface is where thegap forms. As the gap opens, a burst of unscreened electricfield either plunges inside the hole or moves outwards. Sub-sequent pair creation occurs in this burst as it propagates,populating the magnetosphere with pair plasma. This is vis-ible in the upper panel in Fig. 2, which shows a spacetimediagram of the pair creation rate. This highlights the vari-ability of the gap as well as its small spatial extent.A typical sequence of bursts from the high opacity simula-tions is shown in Fig. 5. The electrostatic gap that opens ac-celerates particles, which produce high-energy photons thatsoon pair produce high-energy particles. As these secondaryparticles are created, they gradually screen the electric fieldparallel to the field lines. The intensity and duration of thebursts are highly variable. They have a spatial extent of aFigure 2: Top panel: spacetime diagram of the pair creationrate at θ = π/ θ = π/ r g (see Fig. 5), which appears promising for in-terpreting ultra-fast variability of AGN. We find that the gapsize is controlled by the IC mean free path. At high opac-ity, the gap width is comparable to the IC mean free path inthe Thomson regime r g /τ . The gap width, measured withthe unscreened electric field, is ∼ . r g at τ =
30. At lowopacity, the mean free path becomes comparable to r g . Par-ticles reach high Lorentz factors in the gap, so the IC crosssection drops, further increasing the mean free path.The multiplicity of the plasma flow is high in the gap(around 10), and reaches 2 outside of a burst. The highopacity solution is already very close to being force-free.We observed that the whole magnetosphere, despite beingtime-dependent due to the bursts, rotates consistently at theoptimal predicted angular velocity ≈ ω BH / B ˜ ε wouldlikely increase the multiplicity and allow the magnetosphereto be even more force-free. The total Poynting power outputmeasured in the simulations is also consistent with the BZprediction [5, 33] L BZ = B ω / γγ pairproduction processes can supply su ffi cient plasma to activatethis mechanism.At low opacity a sizeable fraction of the Poynting flux(around 20%) is dissipated within the numerical box. Alarge fraction of the dissipated energy goes into high-energyphotons and leptons. The bulk energy-at-infinity of the lep-tons within the ergosphere can be negative, as emphasizedin P19; we find that they significantly contribute to energyextraction from the black hole at low opacity. At higheropacity dissipation is smaller since the gap is narrow. Theenergy flux carried by leptons becomes negligible . Thedissipated energy is rather deposited in photons below thepair creation threshold, which we remove from the simula-tion to save computing time. The power carried by thesephotons can be estimated by computing the dissipation rate (cid:82) V E i J i d V integrated over the whole simulation box. Athigh optical depths, the dissipated power is around 3% ofthe output Poynting flux. Therefore these bursts are likelyto come with gamma-ray emission, possibly detectable fromEarth.Our results show some similarities with 1D models, butalso important di ff erences which justify the need for multi-dimensional simulations. Similarly to Chen and Yuan [12],we find that the gap opens quasi-periodically. However, un-like them we find that discharges happen at the inner light This does not contradict the conclusion, obtained in P19, that particleswith negative energy-at-infinity can contribute significantly to black-holeenergy extraction. In their study, most of them were located in a currentsheet, while there is none in our simulations.
Figure 3: Snapshot of the phase space for electrons (blackdots) and positrons (red triangles) sampled at θ = π/ ± .
02 during a burst, for τ =
30. Particles aredenoted by sgn( v r )( Γ − Γ is the FIDO-measuredLorentz factor and v r is their radial 3-velocity. The bluesolid line is the normalized unscreened electric field profileat θ . The vertical dash-dotted line marks the location of thelight surface at θ . For clarity, only 20% of the particles aredisplayed.surface, whereas the null surface seems to play no role. Ad-ditionally, while their gap has a size (cid:38) r g , we find that thegap size is much smaller than the black hole size in the highoptical depth regime (although it remains much larger thanthe plasma skin depth). A major di ff erence between 1D and2D is that field lines do not all behave as a coherent entity. Therefore the pair creation bursts have a smaller spatial ex-tent and the time variability is higher in our simulations thanin 1D models. On the other hand, we do not observe thequasi-steady, noisy state obtained by Levinson and Cerutti[11], or by Chen and Yuan [12] at low resolution. This mightbe because field lines can still weakly interact through theelectric field in the ( θ, ϕ ) plane, retaining some coherence atsmall scale.In a future work we will aim to reproduce radio andgamma-ray observations of AGN, by applying the self-consistent radiative transfer treatment used in this study toother magnetic configurations. Although the structure of theoutflow might be quantitatively di ff erent, the inner light sur-face is not expected to depend significantly on the large-scalemagnetic configuration and therefore the broad conclusionswe draw from this study should hold generally. ACKNOWLEDGMENTS
The authors would like to thank A. Levinson, M.Medvedev, V. Beskin and E. Quataert for useful discussions.This work has been supported by the Programme Nationaldes Hautes Énergies of CNRS / INSU, CNES, the France-Berkeley Fund (Project
Multi-dimensional simulations of ergospheric pair discharges around black holes:Supplemental Material
I. MAGNETIC CONFIGURATION
The initial electromagnetic field in our simulations is prescribed by the following 4-potential [17], written in Kerr-Schildspherical coordinates ( t , r , θ, ϕ ): A µ = B r g (cid:32) a cos θ ( r / r g ) + a cos θ , , , − ( r / r g ) + a ( r / r g ) + a cos θ cos θ (cid:33) , (1)where a ∈ [0 ,
1[ is the dimensionless spin parameter of the black hole, r g the gravitational radius and B the strength ofthe magnetic field. Eq. (1) describes a solution to Maxwell’s equations for a black hole with a magnetic monopole. Theelectromagnetic fields are then derived from A µ : B r = √ h ∂ θ A ϕ , (2) B θ = − √ h ∂ r A ϕ , (3) B ϕ = , (4) E r = ∂ r A t , (5) E θ = ∂ θ A t , (6) E ϕ = ∂ ϕ A t = , (7)where h is the determinant of the spatial 3-metric. We verified that the vacuum electromagnetic field relaxes to the solutiondescribed by Eq. (1) if we start with a purely radial magnetic field and no electric field. II. RADIATIVE TRANSFER
We include two radiative processes in our model: inverse Compton (IC) scattering and photon-photon pair production.We introduce “fiducial observers” (FIDOs), whose wordlines are orthogonal to spatial hypersurfaces. We take advantage ofthe fact that FIDOs are locally inertial observers, so that the laws of special relativity can be applied, provided we only useFIDO-measured physical quantities. For simplicity, we assume the soft background radiation field to be isotropic, mono-energetic, and uniform, with density n in the FIDOs’ rest frame. We also neglect pairs that would be produced by theannihilation of the soft background radiation field on itself, i.e. due to MeV emission from the radiatively-ine ffi cient flow.The density of pairs created through this process is usually expected to be much smaller than the Goldreich-Julian density,and therefore too low to screen the gap, for the very low accretion rate found for M87* [6, 27]. A. Condition for interaction
The opacity of IC scattering, for a lepton of Lorentz factor γ = (1 − β ) − / propagating in the soft radiation field, iscomputed as [18] κ IC ( γ ) = τ r g (cid:90) π − π d θ sin θ (1 − β cos θ ) σ KN ( ε , γ, θ ) , (8)where σ KN is the Klein-Nishina cross section, and τ = n r g σ T is the fiducial opacity ( σ T is the Thomson cross section).The pair production opacity is computed similarly, using the pair production cross-section σ γγ instead of σ KN . The opticaldepth traversed by a particle whose spatial coordinates have changed by an amount d x i is measured during a time step as δτ = κ (cid:113) h i j d x i d x j , (9)where h i j is the spatial 3-metric. A number p is randomly drawn with uniform probability between 0 and 1; a scatteringevent occurs provided p < − exp( − δτ ). B. Inverse Compton scattering
We consider, in the FIDO frame, a lepton of energy γ m e c interacting with a soft photon of energy ε . The photon makesangles ( θ , ϕ ) with the lepton velocity. In the following, quantities defined in the lepton rest frame will be primed. After thescattering, the photon energy is ε . The energy of the photon in the lepton rest frame is ε (cid:48) = ε γ (1 − βµ ), where µ = cos θ and β = (1 − γ − ) / . The kinematics of IC scattering yield ε (cid:48) = ε (cid:48) + ε (cid:48) m e c (1 − cos Θ (cid:48) ) , (10)where cos Θ (cid:48) = µ (cid:48) µ (cid:48) + (cid:113) − µ (cid:48) (cid:113) − µ (cid:48) cos ( ϕ (cid:48) − ϕ (cid:48) ), Θ (cid:48) being the angle between the incoming and the scattered photondirections in the lepton rest frame. We assume that the lepton is very energetic ( γ (cid:29) µ (cid:48) ≈ − Θ (cid:48) ≈ − µ (cid:48) . The energy of the scattered photon in the lepton rest frame ε (cid:48) is determined using the full IC di ff erential cross-section from quantum electrodynamics (QED). Given ε (cid:48) , the scatteringangle in this frame is deduced using Eq. (10): µ (cid:48) = m e c ε (cid:48) − m e c ε (cid:48) − . (11)Finally, another Lorentz transformation gives the energy of the scattered photon back in the FIDO frame: ε = γ (1 + βµ (cid:48) ) ε (cid:48) . (12)Thus, once the angle of the incoming photon θ is randomly drawn, we only need to draw the energy of the scattered photonin the lepton rest frame ε (cid:48) from QED.We can now summarize our Monte-Carlo scheme for IC scattering. First the FIDO-measured Lorentz factor γ = (cid:113) + h jk u j u k of a lepton is computed. Since the radiation field is isotropic, we draw uniformly the random variable µ ∈ [ − ,
1] and use it to compute ε (cid:48) by a Lorentz transformation. The scattered photon energy ε (cid:48) is then determinedusing the full IC di ff erential cross-section. Then µ (cid:48) is given by Eq. (11), and we deduce the energy of the scattered photonin the FIDO frame with Eq. (12). In the code, we create a high-energy photon at the location of the scattering lepton, withenergy ε . Assuming strong relativistic beaming again, the direction of the scattered photon in the FIDO frame is the same asthat of the scattering lepton. The new Lorentz factor of the lepton is γ = γ + ε / m e c − ε / m e c , using energy conservation.Jones [19] derived the analytical photon spectrum scattered by a single lepton bathed in a uniform, isotropic and mono-energetic radiation field, which is valid both in the Thomson and the Klein-Nishina regimes. We confirmed that the photonspectrum obtained in our numerical simulations matched this analytical prediction in both regimes. C. Pair production
We consider two photons of energies ε and ε in the FIDO frame, colliding with an angle θ . In the following, we willassume that ε (cid:28) ε , where ε is the energy of a soft photon from the background radiation field. An electron / positron ( e ± )pair can only be created provided [20] s = ε ε (1 − cos θ ) ≥ ( m e c ) . (13)In the following, quantities defined in the center-of-mass (COM) frame of the pair will be primed. In the limit ε (cid:29) ε , theLorentz factor and velocity of the COM frame with respect to the FIDO frame are γ CM ≈ ε / √ s and β CM = − s /ε [21].The electron and the positron both have the same energy γ (cid:48) m e c = √ s in the COM frame. The angle θ (cid:48) , at which theproduced pair propagates with respect to the gamma-ray direction in the COM frame, is determined by QED. Once µ (cid:48) = cos θ (cid:48) is known, the energy of the electron is given by a Lorentz transformation back to the FIDO frame: γ − = γ CM ( γ (cid:48) + β CM µ (cid:48) (cid:113) γ (cid:48) − , (14)whereas the positron energy is determined by energy conservation: γ + m e c = ε + ε − γ − m e c ≈ ε − γ − m e c . (15)Since the energy distribution is symmetric with respect to ε /
2, we arbitrarily choose to pick the electron first.We can summarize our Monte-Carlo scheme for pair production. First the FIDO-measured energy of a gamma photon ε = (cid:113) h jk u j u k is computed. We draw uniformly the random variable µ = cos θ ∈ [ − ,
1] and use it to compute s fromEq. (13). If s ≤ ( m e c ) then no pair is created. Otherwise we compute γ CM , β CM , γ (cid:48) , and then draw µ (cid:48) using the QEDdi ff erential cross-section for pair creation. The gamma photon is discarded from our simulation, and an e ± pair is created inits place, with the energies of the electron and the positron given respectively by Eq. (14) and (15). We take the direction ofpropagation of the created pair to be along that of the primary gamma-ray. This approximation is valid provided γ CM (cid:29) ε (cid:29) ε .Aharonian et al. [22] derived the analytical pair spectrum for a high-energy photon propagating in an isotropic and mono-energetic radiation field, in the case where the high-energy photon has an energy much greater than that of a photon fromthe background field. We verified that the agreement between this analytical prediction and the output of the algorithm isgood, both close to the pair creation threshold ( s ≈ γ + ≈ ε or γ − ≈ ε ), where the pair’s energy is asymmetric. III. POYNTING FLUX
Fig. 4a shows the total Poynting flux through spheres centered on the black hole, as a function of the radius of that sphere.The fluxes are normalized with the total power output of the black hole predicted by the BZ mechanism [5]: L BZ = B ω , (16)where ω BH = ( ca / r g ) / (1 + √ − a ) is the angular velocity of the black hole. This expression is accurate to second orderin ω BH [33]. The Poynting flux decreases with increasing r because some energy is dissipated in the gap and converted intolepton kinetic energy. Dissipation of the Poynting flux is larger at lower opacity since the non-ideal gap region is wider.Fig. 4b shows the energy-at-infinity fluxes carried by electrons, positrons and high-energy photons (above the pair creationthreshold) in the high-opacity simulation. Their contribution to black-hole energy extraction is very small. At high opacity,the dissipated electromagnetic energy is mostly transferred to photons below the pair creation threshold. (a) r/r g . . . . . . L e m ( r ) / L B Z τ = 10 τ = 20 τ = 30 (b) r/r g − . − . − . . . . . L p a r t ( r ) / L B Z τ = 30 ElectronsPositronsPhotons Figure 4: (a) Steady-state Poynting flux through spherical shells centered on the black hole for three optical depths, τ = τ =
30. All fluxes are scaled with L BZ . IV. ANGULAR VELOCITY OF THE FIELD LINES
The field lines’ angular velocity can be evaluated as [5, 23] Ω F = − E θ / √ hB r . Fig. 5 shows that the whole magnetosphererotates consistently at Ω F ≈ ω BH / τ = r/r g . . . Ω F / ω h τ = 5 τ = 20 τ = 30 Figure 5: Angular velocity of the field lines, averaged over θ , as a function of r , for three optical depths, τ =
5, 20 and 30. [1] R. C. Walker, P. E. Hardee, F. B. Davies, C. Ly, and W. Junor, Astrophys. J. , 128 (2018), arXiv:1802.06166 [astro-ph.HE].[2] A. Abramowski, F. Acero, F. Aharonian, A. G. Akhperjanian, G. Anton, A. Balzer, A. Barnacka, U. Barres de Almeida, Y. Becherini,J. Becker, et al. , Astrophys. J. , 151 (2012), arXiv:1111.5341 [astro-ph.CO].[3] F. Aharonian, A. G. Akhperjanian, A. R. Bazer-Bachi, M. Beilicke, W. Benbow, D. Berge, K. Bernlöhr, C. Boisson, O. Bolz,V. Borrel, et al. , Science , 1424 (2006), astro-ph / et al. , Astrophys. J. , L1 (2019), arXiv:1906.11238.[5] R. D. Blandford and R. L. Znajek, mnras , 433 (1977).[6] A. Levinson and F. Rieger, Astrophys. J. , 123 (2011), arXiv:1011.5319 [astro-ph.HE].[7] A. E. Broderick and A. Tchekhovskoy, Astrophys. J. , 97 (2015), arXiv:1506.04754 [astro-ph.HE].[8] K. Hirotani and H.-Y. Pu, Astrophys. J. , 50 (2016), arXiv:1512.05026 [astro-ph.HE].[9] A. Levinson and N. Segev, Phys. Rev. D , 123006 (2017), arXiv:1709.09397 [astro-ph.HE].[10] S. S. Komissarov, mnras , 1431 (2004), astro-ph / , A184 (2018), arXiv:1803.04427 [astro-ph.HE].[12] A. Y. Chen and Y. Yuan, arXiv e-prints (2019), arXiv:1908.06919 [astro-ph.HE].[13] K. Parfrey, A. Philippov, and B. Cerutti, Physical Review Letters , 035101 (2019), arXiv:1810.03613 [astro-ph.HE].[14] B. Cerutti and G. Werner, “Zeltron: Explicit 3D relativistic electromagnetic Particle-In-Cell code,” (2019), ascl:1911.012.[15] B. Cerutti, G. R. Werner, D. A. Uzdensky, and M. C. Begelman, Astrophys. J. , 147 (2013), arXiv:1302.6247 [astro-ph.HE].[16] B. Cerutti, A. Philippov, K. Parfrey, and A. Spitkovsky, mnras , 606 (2015), arXiv:1410.3757 [astro-ph.HE].[17] V. P. Frolov and I. D. Novikov, Black Hole Physics: Basic Concepts and New Developments , 1st ed. (Springer, 1998).[18] G. R. Blumenthal and R. J. Gould, Reviews of Modern Physics , 237 (1970).[19] F. C. Jones, Physical Review , 1159 (1968).[20] R. J. Gould and G. P. Schréder, Physical Review , 1404 (1967).[21] S. Bonometto and M. J. Rees, mnras , 21 (1971).[22] F. A. Aharonian, A. M. Atoian, and A. M. Nagapetian, Astrofizika , 323 (1983).[23] S. S. Komissarov and J. C. McKinney, mnras , L49 (2007), astro-ph / , 85 (2007), arXiv:0704.3282 [astro-ph]. [25] Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball,M. Balokovi´c, J. Barrett, et al. , Astrophys. J. , L5 (2019), arXiv:1906.11242.[26] A. A. Abdo, M. Ackermann, M. Ajello, W. B. Atwood, M. Axelsson, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol, et al. , Astrophys. J. , 55 (2009), arXiv:0910.3565 [astro-ph.HE].[27] M. Mo´scibrodzka, C. F. Gammie, J. C. Dolence, and H. Shiokawa, Astrophys. J. , 9 (2011), arXiv:1104.2042 [astro-ph.HE].[28] P. Goldreich and W. H. Julian, Astrophys. J. , 869 (1969).[29] A. N. Timokhin and A. K. Harding, Astrophys. J. , 12 (2019), arXiv:1803.08924 [astro-ph.HE].[30] S. S. Komissarov, mnras , 427 (2004).[31] A. Nathanail and I. Contopoulos, Astrophys. J. , 186 (2014), arXiv:1404.0549 [astro-ph.HE].[32] M. Takahashi, S. Nitta, Y. Tatematsu, and A. Tomimatsu, Astrophys. J. , 206 (1990).[33] A. Tchekhovskoy, R. Narayan, and J. C. McKinney, Astrophys. J.711