Centrifugal acceleration of protons by a supermassive black hole
aa r X i v : . [ a s t r o - ph . H E ] J a n MNRAS , 1– ?? (2019) Preprint 13 января 2020 г. Compiled using MNRAS L A TEX style file v3.0
Centrifugal acceleration of protons by supermassive blackhole
Istomin Ya. N. , ⋆ , Gunya A. A. † ID P.N. Lebedev Physical Institute, Leninsky Prospect 53, Moscow 119991, Russia ‡ Moscow Institute Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow region, 141700, Russia
ABSTRACT
The centrifugal acceleration is due to the rotating poloidal magnetic field in themagnetosphere creates the electric field which is orthogonal to the magnetic field.Charged particles with finite cyclotron radii can move along the electric field andreceive energy. Centrifugal acceleration pushes particles to the periphery, where theirazimuthal velocity reaches the light speed. We have calculated particle trajectories bynumerical and analytical methods. The maximum obtained energies depend on theparameter of the particle magnetization κ , which is the ratio of rotation frequency ofmagnetic field lines in the magnetosphere Ω F to non-relativistic cyclotron frequency ofparticles ω c , κ = Ω F /ω c << , and from the parameter α which is the ratio of toroidalmagnetic field B T to the poloidal one B P , α = B T /B P . It is shown that for smalltoroidal fields, α < κ / , the maximum Lorentz factor γ m is only the square rootof magnetization, γ m = κ − / , while for large toroidal fields, α > κ / , the energyincreases significantly, γ m = κ − / . However, the maximum possible acceleration, γ m = κ − , is not achieved in the magnetosphere. For a number of active galacticnuclei, such as M87, maximum values of Lorentz factor for accelerated protons arefound. Also for special case of Sgr. A* estimations of the maximum proton energy andits energy flux are obtained. They are in agreement with experimental data obtainedby HESS Cherenkov telescope. Key words: particle acceleration, active galactic nuclei, black hole
At present, it is believed that the most powerful sources ofgamma radiation, such as relativistic jets of blazars, Lac,quasars and radio galaxies, are the most efficient particleaccelerators (Blandford, Meier and Readhead, (2019)).Following the classic paradigm of active galactic nuclei(AGN) (Robson, (1996)), the base of these objects containsa central object in the form of a supermassive blackhole surrounded by an accretion disk; its matter accretsonto the surface of the event horizon. For radio galaxies,blazars and Lac, a part of the accreted matter does notfall onto the centre, but leaves along the axis of rotationoutwards, perpendicular to the accretion disk plane, forminga collimated plasma flow (jet). The radiation of jets isnon-thermal, variable and lies in a wide range of thespectrum of electromagnetic waves, from radio to X-ray. A ⋆ E-mail: [email protected] † E-mail: [email protected] ‡ number of Cherenkov astronomy observatories of (such asVERITAS, HESS, MAGIC) over the past few years haveregistered sources of gamma rays of extremely high energiesup beyond the TeV range. The most striking sources of suchenergy are M87, NGC 5128, 1ES 2344+514. In addition toextragalactic sources, a significant value of radiation energy(up to eV ) was discovered in Sgr. A*, the centre of ourGalaxy (Abramowski et al., 2016).The main carriers of such high energies are bothelectrons with energies up to TeV units (Ghisellini et al.,1998) and protons with energies up to ≃ eV. Toexplain the origin of particles with energies in this range,the centrifugal acceleration mechanism was proposed as themain acceleration mechanism (Rieger and Mannheim, 2000);(Rieger and Aharonian, 2008). Many works consideredthe acceleration of electrons. However, much of electrons’energy is lost are due to synchrotron radiation, so theiracceleration is not as efficient as the acceleration ofprotons. We are interested in the possibility of reachingthe limiting energy achieved during the acceleration ofprotons, the main component of the cosmic ray spectrum c (cid:13) Istomin Ya. N., Gunya A. A. (Ginzburg, 1957). The acceleration of particles in theAGN can occur both in the accretion disk and in theblack hole magnetosphere (Istomin and Sol, 2009). Theacceleration in the disk is due to the presence of a turbulentelectromagnetic field generated by the turbulent motion ofthe matter of the accretion disk, which leads to accretion.In the magnetosphere, acceleration of charged particles isassociated with the presence of an electric field proportionalto the angular velocity of rotation of the magnetosphericplasma that is brought into the rotation by the rotatingblack hole (Blandford and Znajek, 1977), and its accretiondisk (Blandford and Payne, 1982). Also, acceleration canbe induced by shock waves and in regions of reconnectionof magnetic field lines. However, the first two mechanismsare always present and are stationary or quasi-stationary,while, in general, the processes leading to the appearanceof shock waves and regions with a reverse magnetic fieldmay not appear. Here we consider only the acceleration ofcharged particles in the magnetosphere of a black hole. Theacceleration of particles in the plasma disk by a stochasticelectromagnetic field was studied in detail by Istomin andSol, (2009). Due to interactions of energetic particles withthe photon field of the disk (the disk temperature is about 10eV), the maximum Lorentz factor of protons cannot reachlarge values. In this case, the acceleration in the disk canbe considered as the initial process of pre-acceleration ofparticles injected from disk to the magnetosphere. This workis this continuation of part of the work started by Istominand Sol, (2009), where the acceleration of particles in ablack hole magnetosphere was considered in the vicinity theaccretion disk, neglecting the toroidal magnetic field.The magnetic sield structure is described in section 2.The equations for charged particles motion are given insection 3, along with the approximate analytical expressionfor maximum energies. Section 4 presense calculations fora range of real AGNs in terms of proton acceleration. Theresults are discussed in section 5.
Due to rather high temperature ( T ≃ eV ) and therelatively low plasma density of the disk ( n ≤ cm − )the plasma fluid can be described by equations of idealmagnetic hydrodynamics (MHD). This is due to the factthat particles (protons and electrons) are magnetized, i.e.the cyclotron frequency of protons in a magnetic field B , ω ci ≃ ( B/ G ) s − much larger than frequency of proton-proton collisions ν ≃ − ( n/ cm − )( T / eV ) − / s − .Therefore, the electric field in the disk E is connected withthe magnetic field B by the relation E = − c u × B , (1)where u is the velocity of the disk plasma. For the thin diskit means that E φ = u ρ B z . Here E φ is the toroidal electricfield, u ρ is the radial velocity of the disk matter, which doesnot equal to zero during accretion, B z is the componentof the magnetic field orthogonal to the disk plane. Thetoroidal electric field E φ for stationary or quasistationaryaccretion is zero as the electric field is potential. Therefore,there should be no vertical magnetic field in the disk. Thedisk plasma should be polarized so as to push out the B z component. Therefore at the boundary of the accretion diskthe magnetic field in the magnetosphere can only have aradial, along with a spherical radius r , component B r andthe toroidal magnetic field B φ . By virtue of axial symmetry,i.e. non-dependence of all quantities on the azimuth angle φ , the radial field B r ∝ r − . It is the monopoly field. Butdue to divergence-free, the radial magnetic field must havedifferent signs in different half spaces, z > and z < .Thus, B r ∝ s/r , where the value of s is the sign of z , s =sign( z ) . This configuration of the magnetic field is called thesplit monopoly. A toroidal field B φ , superimposed onto theradial magnetic field, is created by radial electric currents j r ( j φ = j θ = 0) , flowing along the radial magnetic field inthe disk and in the magnetosphere in the immediate vicinityof the black hole. The forward and reverse currents areclosed in the jet along the axis of rotation of the black hole,perpendicular to the accretion disk. A toroidal magneticfield in the general case has the form B φ ∝ r − F ( θ ) ,where F ( θ ) is some function of the polar angle θ . It isdetermined by the dependence of the radial current j r ( θ ) on the polar angle, i.e. how the jets electric current is closedin the black hole vicinity, j r ∝ ∂ (sin θB φ ) /∂θ/r sin θ . In thesimplest case, F ( θ ) = const ( θ ) . Since the electric currentsof the jet above the disk and under the disk run in differentdirections, the toroidal field, as well as the poloidal field, isproportional to the value of s , B φ ∝ s/r . The configurationof magnetic field lines, defined by the ratio dr/B r = rdφ/B φ ,is expanding spirals r = r φ + r , where the values of r and r are constants. Figure 1 shows the magnetic field lines ofthis configuration in the magnetosphere.It should be noted that the absence of the verticalmagnetic field B z in the disk refers to a stationary, orquasi-stationary, magnetic field. It means that due to themagnetization of the plasma, charged particles (protons andelectrons) cannot pass through vertical magnetic field duringaccretion. However, if electric currents in the disk do nothave time to change so that B z ≃ , then a non-zero verticalmagnetic field will be captured by the accreating plasma fluxgoing to the centre and be amplified due to the magnetic fluxconservation. This can lead to a decrease of the accretionrate, up to its cessation. Also can appear of non-stationaryelectromagnetic fields in the disk and particle acceleration.However, numerical calculations of the structure of disksand their surroundings near the black hole using generalrelativity magneto hydro dynamics (GRMHD) show thatarising inside the disk vortices strongly elongated along thedisk with mainly horizontal magnetic field (Mishra et al.,2016). This means that indeed, although the field is notstationary, the electric currents in the plasma of the diskare designed so that the magnetic field created by themdoes not prevent accretion. For this, it is necessary that theplasma conductivity σ c be small enough to suppress shortvortices of the order of the disk thickness H , in which thevertical component of the magnetic field is large, σ c < v/H .Here the velocity v is the velocity of matter in a vortex.The vortex revolution time is ≃ H/v . Apparently, thenumerical magnetic viscosity (inversely to the conductivity)in the calculations satisfies this condition. Also, the realconductivity of the turbulent matter is anomalous andsignificantly less than the classical one. Because of this, themagnetic lines above the disk are directed tangentially to thedisk (Morales Teixeira, Avara, & McKinney, 2018) and the
MNRAS , 1– ?? (2019) cceleration of protons by black holes Figure 1.
Scheme of picture of magnetic field lines in themagnetosphere in the vicinity of a rotating black hole poloidal magnetic field in magnetosphere is close to radialone due to its strong compression by the plasma pressureof the disk near the central object (Kathirgamaraju et al.,2019; Liska et al., 2019).For split monopole magnetic field lines directly exitfrom the vicinity of the central black hole. They are involvedinto the rotation by the black hole and rotate with anangular velocity of Ω F , which for optimal matching is halfthe angular velocity of rotation of a black hole Ω H , Ω F =Ω H / (Blandford and Znajek, 1977). The rotating radialmagnetic field excites in the magnetosphere a θ componentof the electric field, E θ , which, by virtue of Faraday’s law, is E θ dl = − c d Φ dt = − r sin θB r c dφdt dl. (2)Here the element of length dl is taken along the θ direction.We get E θ = − r Ω F sin θB r c . (3)The relation (3) looks like a condition (1) in ideal MHD,where instead of the fluid velocity u the azimuthal rotationvelocity appears, u = e φ r sin θ Ω F . The electric field (3) inthe magnetosphere arises near the black hole horizon dueto its rotation and is transmitted along the radial magneticfield to the magnetosphere, which is located from the centreright up to the light cylinder surface r = c/ Ω F sin θ . Thepolar field E θ in the magnetosphere is the agent thattransfers the black hole rotation to the rotation of plasma ofthe magnetosphere and, when charges are displaced in thepolar direction, it transfers energy to them. We emphasizehere that the polar electric field E θ (3) is the result thecentral object rotation, and is not a consequence of MHD,which may not be satisfied in the magnetosphere. This topicis discussed in the section ’Discussion’.Thus, in the magnetosphere there is an electromagneticfield with components B r = sB (cid:18) rr L (cid:19) − ,B φ = αsB sin θ (cid:18) rr L (cid:19) − , (4) E θ = − sB sin θ (cid:18) rr L (cid:19) − . Here, r L = c/ Ω F is the natural size of the magnetosphere in the transverse direction. This is the radius of the lightcylinder, at which the electric field is compared with themagnetic one. For a spherical distance r , the light surface isat the distance r = r L / sin θ . The value of B is the value ofthe radial magnetic field at the distance r = r L . At the lightcylinder the value of the radial field falls from the value of B near the accretion disk ( θ ≃ π/ to B sin θ . The coefficient α is the ratio of the toroidal magnetic field to the poloidalone at the distance r = r L from the centre. Since the splitmonopole field and the toroidal field differently depend onthe radial distance r , then at ’small’ distances r < α − r L the poloidal magnetic field prevails, while at ’large’ distances r > α − r L the field becomes predominantly toroidal. So for α > the magnetic field near the light surface, r = r L / sin θ ,where, as we will see, the main acceleration of chargedparticles occurs, the field is mainly toroidal. For small valuesof α , α < , the field is only at large distances from thecentre, r > ( α sin θ ) − r L , is close to the toroidal one, whilenear the field centre it is close to the poloidal one. Motion of particles of mass m and charge q in theelectromagnetic field of the black hole magnetosphere isdescribed by equations d p dt = q (cid:18) E + 1 c [ v , B ] (cid:19) ,d r dt = v = p mγ , (5) γ = 1 + p m c . Here r and p are the coordinate and the momentum ofa particle, γ is its Lorentz factor. It is convenient for usto introduce dimensionless time, coordinates, velocity andmomentum, t ′ = ω c tγ i , r ′ = r r L , v ′ = v c , p ′ = p mcγ i . (6)The initial value of the Lorentz factor is γ i , thenonrelativistic cyclotron frequency of a particle rotation inthe B field is ω c = qB /mc . Let us also introduce the valueof the Lorentz factor relative to the initial energy, γ ′ = γ/γ i .In these variables, the equations of motion (5) in sphericalcoordinates r, θ, φ (primes are omitted) have the form dp r dt = κrγ (cid:0) p θ + p φ (cid:1) + sαrγ p θ ,dp θ dt = − κrγ (cid:0) p r p θ − p φ cot θ (cid:1) − sr sin θ + sr γ p φ − sαrγ p r ,dp φ dt = − κrγ ( p r + p θ cot θ ) p φ − sr γ p θ , (7) drdt = κγ p r ,dθdt = κrγ p θ . The equations (7) contain two dimensionless constants, α and κ . The value of α , introduced in the previous section,is equal to the ratio of the toroidal magnetic field to the MNRAS , 1– ????
Scheme of picture of magnetic field lines in themagnetosphere in the vicinity of a rotating black hole poloidal magnetic field in magnetosphere is close to radialone due to its strong compression by the plasma pressureof the disk near the central object (Kathirgamaraju et al.,2019; Liska et al., 2019).For split monopole magnetic field lines directly exitfrom the vicinity of the central black hole. They are involvedinto the rotation by the black hole and rotate with anangular velocity of Ω F , which for optimal matching is halfthe angular velocity of rotation of a black hole Ω H , Ω F =Ω H / (Blandford and Znajek, 1977). The rotating radialmagnetic field excites in the magnetosphere a θ componentof the electric field, E θ , which, by virtue of Faraday’s law, is E θ dl = − c d Φ dt = − r sin θB r c dφdt dl. (2)Here the element of length dl is taken along the θ direction.We get E θ = − r Ω F sin θB r c . (3)The relation (3) looks like a condition (1) in ideal MHD,where instead of the fluid velocity u the azimuthal rotationvelocity appears, u = e φ r sin θ Ω F . The electric field (3) inthe magnetosphere arises near the black hole horizon dueto its rotation and is transmitted along the radial magneticfield to the magnetosphere, which is located from the centreright up to the light cylinder surface r = c/ Ω F sin θ . Thepolar field E θ in the magnetosphere is the agent thattransfers the black hole rotation to the rotation of plasma ofthe magnetosphere and, when charges are displaced in thepolar direction, it transfers energy to them. We emphasizehere that the polar electric field E θ (3) is the result thecentral object rotation, and is not a consequence of MHD,which may not be satisfied in the magnetosphere. This topicis discussed in the section ’Discussion’.Thus, in the magnetosphere there is an electromagneticfield with components B r = sB (cid:18) rr L (cid:19) − ,B φ = αsB sin θ (cid:18) rr L (cid:19) − , (4) E θ = − sB sin θ (cid:18) rr L (cid:19) − . Here, r L = c/ Ω F is the natural size of the magnetosphere in the transverse direction. This is the radius of the lightcylinder, at which the electric field is compared with themagnetic one. For a spherical distance r , the light surface isat the distance r = r L / sin θ . The value of B is the value ofthe radial magnetic field at the distance r = r L . At the lightcylinder the value of the radial field falls from the value of B near the accretion disk ( θ ≃ π/ to B sin θ . The coefficient α is the ratio of the toroidal magnetic field to the poloidalone at the distance r = r L from the centre. Since the splitmonopole field and the toroidal field differently depend onthe radial distance r , then at ’small’ distances r < α − r L the poloidal magnetic field prevails, while at ’large’ distances r > α − r L the field becomes predominantly toroidal. So for α > the magnetic field near the light surface, r = r L / sin θ ,where, as we will see, the main acceleration of chargedparticles occurs, the field is mainly toroidal. For small valuesof α , α < , the field is only at large distances from thecentre, r > ( α sin θ ) − r L , is close to the toroidal one, whilenear the field centre it is close to the poloidal one. Motion of particles of mass m and charge q in theelectromagnetic field of the black hole magnetosphere isdescribed by equations d p dt = q (cid:18) E + 1 c [ v , B ] (cid:19) ,d r dt = v = p mγ , (5) γ = 1 + p m c . Here r and p are the coordinate and the momentum ofa particle, γ is its Lorentz factor. It is convenient for usto introduce dimensionless time, coordinates, velocity andmomentum, t ′ = ω c tγ i , r ′ = r r L , v ′ = v c , p ′ = p mcγ i . (6)The initial value of the Lorentz factor is γ i , thenonrelativistic cyclotron frequency of a particle rotation inthe B field is ω c = qB /mc . Let us also introduce the valueof the Lorentz factor relative to the initial energy, γ ′ = γ/γ i .In these variables, the equations of motion (5) in sphericalcoordinates r, θ, φ (primes are omitted) have the form dp r dt = κrγ (cid:0) p θ + p φ (cid:1) + sαrγ p θ ,dp θ dt = − κrγ (cid:0) p r p θ − p φ cot θ (cid:1) − sr sin θ + sr γ p φ − sαrγ p r ,dp φ dt = − κrγ ( p r + p θ cot θ ) p φ − sr γ p θ , (7) drdt = κγ p r ,dθdt = κrγ p θ . The equations (7) contain two dimensionless constants, α and κ . The value of α , introduced in the previous section,is equal to the ratio of the toroidal magnetic field to the MNRAS , 1– ???? (2019) Istomin Ya. N., Gunya A. A. L i gh t cy li nd e r s u r f ace [ (cid:1) [ t ]] P ϕ [ t ] Figure 2.
The azimuthal momentum of the particle p φ versusthe radial distance r sin θ for κ = 10 − and α = 10 − . The lightcylinder surface is located at 10 on the abscissa axis. Light cylinder surface - - [ Θ [ t ]] P θ [ t ] Figure 3.
The polar momentum of the particle p θ versus theradial distance r sin θ for the same parameters as in Fig. 2. poloidal one at the distance r = r L , α = B φ /B r | r = r L . Thevalue of κ is equal to κ = cγ i r L ω c = r c r L = Ω F ω c /γ i (8)and is actually the parameter of a charged particlemagnetization, that is the ratio the cyclotron radius of aparticle, r c = cγ i /ω c , to the radius of the light cylinder, orthe ratio of the rotation frequency of magnetic field lines tothe cyclotron frequency. It is clear that in the magnetosphereonly magnetized particles can be accelerated, κ < , γ i <γ = r L ω c /c , which conforms the Hillas criterion (Hillas,1984), which is geometric in nature: it is impossible toaccelerate a particle in a magnetic field whose scale is smallerthan the cyclotron radius.The acceleration of ’cold’ particles, γ i ≃ is considered.In this case, κ << . The question is to find the energymaximum γ m when proton reaches the light surface r = r L / sin θ . First of all need to solve a system of equations (7).The numerical calculation of the proton trajectory startingfrom the magnetosphere internal region, r < r L , is show inFigures (2) and (3).First, it can be seen that the behavior of the toroidalmomentum p φ near the light surface, r ≃ / sin θ , practicallydoes not differ from the behavior of the Lorentz factor γ (energy). This is due to the fact that the system (7) has L i gh t cy li nd e r s u r f ace [ Θ [ t ]] γ Figure 4.
The Lorentz factor of the particle γ versus the radialdistance r sin θ for same parameters as in Fig. 2 two integrals of motion: energy E = const and angularmomentum L = const , E = γ − sκ cos θ, L = rp φ sin θ − γ. (9)As well as energy consisting of particle energy and thework of an electric field, the angular momentum is the sumof the mechanical momentum and the momentum of theelectromagnetic field. Since the initial radius r , from whichthe particle starts, is less than unity, r << , then, as itfollows from the conservation of the angular momentum (9), p φ max = γ m − . The value of γ m is equal to the maximumLorentz factor of a particle achieved at the light surface. Inaddition, γ = p r + p θ + p φ + 1 /γ i , and ( p r + p θ ) | max =2 γ m ( γ m >> . Thus, particle acceleration occurs in theazimuthal direction, p φ ≃ γ , while the poloidal componentsof a particle are much smaller, p r ≃ p θ ≃ γ / . This is thecentrifugal acceleration.Secondly, the angle θ varies strongly only in the vicinityof the light surface, practically without changing on the mainpart of the trajectory (Fig. (5)). As we will see, always γ <<κ − , κ << , and it follows from the energy conservationlaw (9) that cos θ − cos θ << . Since the main accelerationoccurs near the light surface (Fig. (4)), then ∆ θ = θ − θ = − σκγ m / sin θ . On the other hand, dθ/dr = p θ sin θ/p r =∆ θ/ ∆ r , where ∆ r is the size of the region near the lightsurface r = 1 / sin θ , where the main acceleration of particlesoccurs ∆ r = − sκγ m sin θ p r p θ | r =1 / sin θ . (10)Small angles θ ≃ are not considered because the possibleregion of acceleration in this case is far from the centre,where the fields are small and the acceleration is notefficient. Passing in the system (7) from time derivativesto derivatives over coordinate r, d/dt = d/dr ( p r /γ ) , andreplacing derivatives by fractions d p /dr ≃ p | r =1 / sin θ / ∆ r ,we obtain an algebraic system of equations that determines MNRAS , 1– ?? (2019) cceleration of protons by black holes Light cylinder surface (cid:3)(cid:4)(cid:5) [ Θ [ t ]] Θ [ t ] Figure 5.
The polar angle versus the radial distance r sin( θ ) for κ = 10 − Θ γ Figure 6.
The Lorentz factor of a particle at the light surface γ m versus the polar angle θ for κ = 10 − . Points are the resultof numerical calculations, the curve is the dependance (12). the values of ( γ m , p ) | r =1 / sin θ at the light surface r = 1 / sin θp r + αγ m sin θ p r − γ m (sin θ + κγ m | cos θ | )sin θ = 0; p θ = − p r sκγ m sin θ sin θ + κγ m | cos θ | ; (11) p θ + p r = 2 γ m . First consider the case of small values of α . Put (11) α = 0 in the system of equations we find γ m = κ − / sin θ. (12)This result coincides with the expression obtained in(Istomin and Sol, 2009) for the case of θ = π/ , wherethe acceleration of particles in the magnetosphere wasconsidered only near the accretion disk. The expression γ m = κ − / (sin θ = 1) was obtained from the analysis ofnumerical calculations of particle trajectories, while here weuse the analytical approximation. We show in Figure (6)the comparison of numerical calculations (points) with theanalytical expression (12). There are in a good agreement.To obtain a general expression for γ m for the arbitraryvalue of α , we need to solve the quadratic equation, thefirst equation of the system (11), with respect to the valueof p r . However, the resulting expression is inconvenient foranalysis, so we will find an approximate expression for thevalue of p r in the case of large values of α, α γ m >> ,p r ≃ sin θ + κγ m | cos θ | α sin θ . (13)The result is γ m ≃ / (cid:0) ακ − (cid:1) / sin / θ. (14) - - - - Log10 [ α ] Log10 [ γ ] Figure 7.
The maximum Lorentz factor of particle γ m versus thevalue of α for κ = 10 − . The slope of the curve for α > − justcorresponds to the dependence γ m ∝ α / (14). By comparing expressions (12) and (14) obtain the valueof α = α , at which the dependence of the maximumLorentz factor γ m on the magnetization parameter κ at smalltoroidal fields, α < α , transits to the dependence at largetoroidal fields, α > α , α = 2 − / κ / sin / θ ≃ κ / . (15)Figure (3) depicts the dependence of γ m on the toroidalmagnetic field magnitude. The dots correspond the valuescalculated numerically using the equations of motion ofparticles in the magnetosphere (7). Clearly visible is thetransition from independent α at small values of α relationto the relation of γ m ∝ α / in the region α > α . Theresult of our studies of particle acceleration shows thatthe acceleration depends not only on the magnitude of themagnetic field - the magnetization parameter κ (8), but alsoon the topology of the magnetic field, on the ratio betweenthe poloidal and toroidal magnetic fields.The acceleration of particles in the magnetosphere ofthe rotating black hole is carried out, by the electric field E θ (4). This field arises when magnetic field lines rotatewith the angular velocity of Ω F , which is proportional tothe angular velocity of a black hole rotation. The particle,making cyclotron rotation, and drifting in crossed fields inthe direction of rotation due to polarization drift (increasingits toroidal momentum) shifts from the original magneticsurface and acquires additional energy in the electric field.The strongest deviation occurs near the light surface at thedistance ∆ r (10) from it. Here, accelerating, the particleincreases its cyclotron radius and accelerates even moreuntil it reaches the light surface. In the toroidal magneticfield near the light surface, the particle, in contrast tomovement in the poloidal field, also experiences centrifugaland gradient drifts directed along the electric field, whichdeviate it from the given magnetic surface and producesmore efficient acceleration. Here the values of γ and γ m arevalues of the Lorentz factor and the maximum Lorentz factorrelative to the value of γ i , which is the initial Lorentz factor.Actually expressions γ m ≃ κ − / = γ / , α < γ − / ; MNRAS , 1– ????
The maximum Lorentz factor of particle γ m versus thevalue of α for κ = 10 − . The slope of the curve for α > − justcorresponds to the dependence γ m ∝ α / (14). By comparing expressions (12) and (14) obtain the valueof α = α , at which the dependence of the maximumLorentz factor γ m on the magnetization parameter κ at smalltoroidal fields, α < α , transits to the dependence at largetoroidal fields, α > α , α = 2 − / κ / sin / θ ≃ κ / . (15)Figure (3) depicts the dependence of γ m on the toroidalmagnetic field magnitude. The dots correspond the valuescalculated numerically using the equations of motion ofparticles in the magnetosphere (7). Clearly visible is thetransition from independent α at small values of α relationto the relation of γ m ∝ α / in the region α > α . Theresult of our studies of particle acceleration shows thatthe acceleration depends not only on the magnitude of themagnetic field - the magnetization parameter κ (8), but alsoon the topology of the magnetic field, on the ratio betweenthe poloidal and toroidal magnetic fields.The acceleration of particles in the magnetosphere ofthe rotating black hole is carried out, by the electric field E θ (4). This field arises when magnetic field lines rotatewith the angular velocity of Ω F , which is proportional tothe angular velocity of a black hole rotation. The particle,making cyclotron rotation, and drifting in crossed fields inthe direction of rotation due to polarization drift (increasingits toroidal momentum) shifts from the original magneticsurface and acquires additional energy in the electric field.The strongest deviation occurs near the light surface at thedistance ∆ r (10) from it. Here, accelerating, the particleincreases its cyclotron radius and accelerates even moreuntil it reaches the light surface. In the toroidal magneticfield near the light surface, the particle, in contrast tomovement in the poloidal field, also experiences centrifugaland gradient drifts directed along the electric field, whichdeviate it from the given magnetic surface and producesmore efficient acceleration. Here the values of γ and γ m arevalues of the Lorentz factor and the maximum Lorentz factorrelative to the value of γ i , which is the initial Lorentz factor.Actually expressions γ m ≃ κ − / = γ / , α < γ − / ; MNRAS , 1– ???? (2019) Istomin Ya. N., Gunya A. A. γ m ≃ κ − / = γ / , α > γ − / , (16)are valid only for the acceleration of ’cold’ particles, γ i ≃ .The value of γ = ω c / Ω F is the maximum possible valueof the energy of accelerated particles (the Hillas criterion).When γ = γ , the cyclotron radius of the particle becomesequal to the radius of the light cylinder. In fact, theacceleration is not so strong, the powers are / and / , not , in the real field of the magnetosphere of the rotating blackhole with an accretion disk. But the presence of the jet, i.e.the toroidal magnetic field generated by an electric currentflowing in a jet, makes acceleration more efficient (the powerof / ). If particles are accelerated not from ’cold’ state, butfrom the Lorentz factor γ i > , then the expressions (16) forthem take the form γ m ≃ ( γ i γ ) / , α < ( γ /γ i ) − / ; γ m ≃ γ / i γ / , α > ( γ /γ i ) − / . (17)If particles are pre-accelerated in the black holemagnetosphere or in a turbulent accretion disk to energies γ i >> , as it was shown by (Istomin and Sol, 2009),then the maximum energy γ m , obtained in the processof centrifugal acceleration, can significantly approach thevalue of γ .The dependence of γ m on the angle θ , see formulas (12,14), means that at different heights z above the equatorialplane the particle energy is also different. Acceleratedparticles, leaving the vicinity of the black hole, will forma single energy spectrum. Summing them, crossing the lightsurface r = r L / sin θ and acquiring energy γ m , that dependson the angle θ , we can obtain the distribution function offast particles f ( γ m ) . As z = r L cot θ, dz = − dθ/ sin θ ,and f ( γ m ) dγ m ∝ πr L ndz , where n is the density of the’primary’ particles in the magnetosphere inside the lightsurface ( n ≃ const ), we get f ( γ m ) ∝ γ − m , γ m < γ / , α < γ − / ; f ( γ m ) ∝ γ − . m , γ m < γ / , α > γ − / . (18)Here we consider that angles θ differ from some smallvalue of θ , determined by the height of the magnetosphere,to θ < π/ . The angle θ separates region of themagnetosphere adjacent to the accretion disk, where thematter rotates with the speed close to the speed of rotationof the disk, and the region where the matter rotates withthe speed close to the speed of rotation of the black hole.Therefore, when getting the expression (18), one can put cos θ ≃ . The value of sin θ basically takes small values,which corresponds to the energies γ m < γ / for smalltoroidal fields, and γ m < γ / for large ones.It should be noted that, as we have seen, for largetoroidal fields, large values of α , particle acceleration is moreefficient than in the absence of a toroidal field. Generallyspeaking, there may exist such magnetic field topologiesfor which it is still possible more efficient acceleration, γ m = γ . However, in case a split monopole magnetiсfields to combined with a toroidal field, the acceleration atlarge toroidal fields does not depend on the strength of thepoloidal field, γ m = ( ακ − ) / . It’s evident the geometry ofthe poloidal field that such topologies, if they exist, shouldrarely be realized. Centrifugal acceleration occurs in the magnetosphere of ablack hole directly in its vicinity of the size of the radius ofthe light cylinder, r ≃ r L = c/ Ω F . For optimal matchingof the rotation of a magnetosphere with the rotation of ablack hole the value of Ω F is Ω F = Ω H / , where Ω H is theangular velocity of rotation of the black hole, Ω H = 2 cr g a − a ) / ,r g = 2 GMc . (19)The value of r g is the gravitational radius of a black hole ofthe M mass, r g = 3 · ( M/M ⊙ ) = 3 · M cm . Here thevalue of M is the mass of a black hole measured in units of M ⊙ . We used masses of black hole M given in referredpapers: in the center of our Galaxy Sgr. A* (Ghez, et al.,2008) and six AGN; NGC 1365 (Fazeli, et al., 2019), NGC4051 (Seifina, Chekhtman & Titarchuk, 2018), NGC 4151(Bentz, et al., 2006), NGC 4486 (M 87) (Broderick, et al.,2015), NGC 6166 (Maggorian, et al., 1998) and NGC 7469(Seifina, Titarchuk & Ugolkova, 2018). See the table.Parameter a characterizes the rotation of a black hole,and is equal to its specific angular momentum, a < . So, Ω F = 10 − aM − / [1 + (1 − a ) / ] s − .The magnetic field strength in the magnetosphere B isone of characteristics of the central black hole, it plays akey role in the process of particle acceleration and underliesthe calculation of γ m . The data presented in the work ofDaly (2019) contain magnetic field values for some of themost famous AGN. The magnetic field strength near thehorizon for six active galaxies is given by the Table 1.The value of B is the magnitude of the magnetic field inunits of G . We see that given values of the magneticfield near the horizon B are less than M − / . This isdue to the fact that the maximum magnetic field can beestimated as follows. Suppose that for a central object withthe Eddington luminosity, L ≃ L Edd , the magnetic fieldenergy density B / π near the horizon is comparable withthe energy density of the accreting matter (equipartition).Then L Edd ≃ πcr g ( B / π ) , and one can introduce thevalue of B Edd = (2 L Edd /cr g ) / ≃ . · M − / G. Thisvalue, on the one hand, is the characteristic value of themagnetic field in AGN and, on the other hand, is its upperlimit, B = bB Edd , b < . The coefficient b depends on themass of the black hole M , on the specific angular momentumof the black hole a , on the kinetic luminosity of the jet L j ,on the bolometric luminosity of the disk L bol and on thevalue of the accretion rate ˙ M . Therefore, to determine b ,it is necessary to know the model of the accretion disk, thejet model, etc. Using physical considerations and empiricaldependences, Daly (2016, 2019) developed a procedure ofdetermining the value of B , as well as a . The Daly’s valuesof the specific angular momentum a for many AGNs turnedout to be close to those estimated by other methods. Thus,the magnetic fields B estimated by him can be consideredquite reliable. Since the cyclotron rotation frequency forprotons is equal to ω c = 10 ( B/ G ) s − , then the value MNRAS , 1– ?? (2019) cceleration of protons by black holes of the maximum possible Lorentz factor γ = ω c / Ω F is γ = 4 · M B − a ) / a ≃ · M B ,a ≃ , (20)which corresponds the proton energy E (1) max = 3 . · M B eV . However, as it follows from our calculations,the split monopole configuration allows to acceleratingparticles to values smaller than γ , γ m = γ / =2 . · ( M B ) / . For protons this is E (2 / max = 2 . · ( M B ) / eV . In the case of a weak toroidal field,acceleration is even less efficient, γ m = γ / = 2 · ( M B ) / , E (1 / max = 1 . · ( M B ) / eV . Here weconsider the example of the galaxy M87 (NGC4486). It hasa powerful jet through which electric current flows. Let usestimate the magnitudes of the poloidal and the toroidalmagnetic fields in its magnetosphere. Jet luminosity is L =10 erg/s (Broderick et al., 2015). Equate it to the electricpower released in the external part of the electric currentloop. With the optimal matching, the resistance of theexternal electrical circuit is equal to the horizon resistance R H = 4 π/c = 377 Ohm Thorne et al., (1986), L = R H I .Then we estimate the value of the electric current I in thejet, I ≃ . · A . This current, flowing in the jet in theforward direction in the centre and in the opposite directionat the periphery, closes in the accretion disk, flowing fromthe outer regions of the disc to the inner region near theblack hole horizon. It creates a toroidal magnetic field B φ in the magnetosphere, B φ ≃ I/ch , where the value of h is half the height of the disk on its inner edge. So, we get B φ ≃ ( M h/r g ) − G . Given h/r g ≃ · − , we find thatthe estimated toroidal magnetic field near the horizon is ofthe order of magnitude given in Table 1 for М87. Let us nowestimate the value of the polodal field B P in the vicinity ofthe horizon. The electric current is generated by the dynamomachine created by a rotating poloidal magnetic field. Thegenerated voltage is U = B P r g Ω H / c (Landau and Lifshitz,1984; Thorne et al., 1986). On the other hand, U = 2 R H I .From here we get B P = 16 πI/cr g = 8 πB φ ( h/r g ) . Thus, theratio of the toroidal magnetic field to the poloidal one, i.e. α , equals α = r g / πh ≃ > . Therefore, for M87 theenergy of accelerated protons corresponds to the estimatedvalue in Table 1.The nucleus of our galaxy Sgr. A* contains the blackhole with the mass of . · M ⊙ . It is not active, itsluminosity is only ≃ − of the Eddington luminosity.There is no jet, and one can think that there is no toroidalmagnetic field, B φ = 0 . Therefore, the maximum Lorentzfactor of particles is given by the expression γ m = γ / ≃ · , which corresponds to the proton energy of E (1 / max ≃ . · eV . However, the observed spectrum of gammaphotons from Sgr. A* is the power law with the exponent ≃− . (Abramowski et al., 2016). It is an intermediate valuebetween the values of -2 and -2.5, given by the expression(18) for weak and strong toroidal magnetic field respectively.It should be assumed that a weak toroidal magnetic field isstill present in the magnetosphere, α ≃ γ − / ≃ . · − .Plasma accretion must be accompanied by the appearanceof an electric current, and hence generation of a toroidalmagnetic field, the way it occurs in case of accretion ontoa neutron star (Istomin and Haensel, 2013). Therefore, the Galaxy M B E (1) max E (2 / max E (1 / max Sgr. A* 0.0043 0.25 . · . · . · Cyg. A* 2.5 0.45 . · . · . · NGC 1365 0.05 5.0 . · . · . · NGC 4051 0.0006 20.0 . · . · . · NGC 4151 0.05 4.0 . · . · . · NGC 4258 0.04 1.45 . · . · . · NGC 4486 6.6 0.07 . · . · . · NGC 6166 28.4 0.11 . · . · . · NGC 7469 0.003 20.0 . · . · . · Table 1.
Upper limits for proton energy. Here, M ≡ M BH / M ⊙ , B ≡ B φ / G , proton energy E (1) max = m p c γ , E (1 / max = m p c γ / , E (2 / max = m p c γ / given in unitsof eV maximum possible energies of accelerated protons lie withininterval E (1 / max < E max < E (2 / max , i.e. . · eV < E max < . · eV , which is consistent with the observationsof the Galactic centre by the HESS Cherenkov telescope(Abramowski et al., 2016).As for the power W radiated into accelerated particles,it is only part of the total power lost by the black hole, S = R H I = U / R H = ( B P / π ) cr L . This part is the ratio ofthe maximum energy of the accelerated protons γ m to themaximum possible particle energy obtained under completetransformation of the energy of rotation of the black holeinto the energy of particles, κ − , that is W = Sγ m κ = Sγ m /γ . For Sgr. A* the value of W lies within . · erg/s < W < · erg/s, since κ − / < γ m < κ − / .Black hole masses have been updated for Sgr A ∗ (Ghezet al. 2008), NGC 1365 (Fazeli et al. 2019), NGC 4051(Seifina et al. 2018), NGC 4151 (Bentz et al. 2006), NGC4258 Gonz´alez-L´opezlira R. A., et al., 2017), NGC 4486(Broderick et al. 2015), NGC 6166 (Maggorian. 1998), NGC7469 (Seifina et al. 2018).It should be noted that we used to estimate the valueof the black hole rotation parameter a ≃ , which istrue for Sgr. A* and M87. For other galaxies, the rotationmay not be so fast, a << . In this case, the size ofthe magnetosphere increases, r L ∝ a − , the magnetizationparameter κ decreases by the same value, which leads toacceleration to higher energies, γ ∝ a − .We would like to emphasize that obtained estimatesof γ m are the Lorentz factor achieved by a ’cold’ particle, γ i ≃ , as a result of acceleration in the magnetosphere. Ifthe particle has undergone preliminary acceleration, γ i > ,then the maximum possible energies increase substantially(16). For example, some particles can be accelerated in aturbulent disk (Istomin and Sol, 2009)).In addition to the above, we would like to add that theresults obtained here can also be used for microquasars suchas SS 433 (Abeysekara, 2018). We have shown that in the magnetosphere of the rotatingblack hole up to the light surface of r < c/ Ω F sin θ ,occurs an acceleration of charged particles. It is boundto the fact that the pole electric field E θ makes possiblethe particle deviation for cyclotron radius of r c = cγ i /ω c MNRAS , 1– ????
Upper limits for proton energy. Here, M ≡ M BH / M ⊙ , B ≡ B φ / G , proton energy E (1) max = m p c γ , E (1 / max = m p c γ / , E (2 / max = m p c γ / given in unitsof eV maximum possible energies of accelerated protons lie withininterval E (1 / max < E max < E (2 / max , i.e. . · eV < E max < . · eV , which is consistent with the observationsof the Galactic centre by the HESS Cherenkov telescope(Abramowski et al., 2016).As for the power W radiated into accelerated particles,it is only part of the total power lost by the black hole, S = R H I = U / R H = ( B P / π ) cr L . This part is the ratio ofthe maximum energy of the accelerated protons γ m to themaximum possible particle energy obtained under completetransformation of the energy of rotation of the black holeinto the energy of particles, κ − , that is W = Sγ m κ = Sγ m /γ . For Sgr. A* the value of W lies within . · erg/s < W < · erg/s, since κ − / < γ m < κ − / .Black hole masses have been updated for Sgr A ∗ (Ghezet al. 2008), NGC 1365 (Fazeli et al. 2019), NGC 4051(Seifina et al. 2018), NGC 4151 (Bentz et al. 2006), NGC4258 Gonz´alez-L´opezlira R. A., et al., 2017), NGC 4486(Broderick et al. 2015), NGC 6166 (Maggorian. 1998), NGC7469 (Seifina et al. 2018).It should be noted that we used to estimate the valueof the black hole rotation parameter a ≃ , which istrue for Sgr. A* and M87. For other galaxies, the rotationmay not be so fast, a << . In this case, the size ofthe magnetosphere increases, r L ∝ a − , the magnetizationparameter κ decreases by the same value, which leads toacceleration to higher energies, γ ∝ a − .We would like to emphasize that obtained estimatesof γ m are the Lorentz factor achieved by a ’cold’ particle, γ i ≃ , as a result of acceleration in the magnetosphere. Ifthe particle has undergone preliminary acceleration, γ i > ,then the maximum possible energies increase substantially(16). For example, some particles can be accelerated in aturbulent disk (Istomin and Sol, 2009)).In addition to the above, we would like to add that theresults obtained here can also be used for microquasars suchas SS 433 (Abeysekara, 2018). We have shown that in the magnetosphere of the rotatingblack hole up to the light surface of r < c/ Ω F sin θ ,occurs an acceleration of charged particles. It is boundto the fact that the pole electric field E θ makes possiblethe particle deviation for cyclotron radius of r c = cγ i /ω c MNRAS , 1– ???? (2019) Istomin Ya. N., Gunya A. A. from an initial magnetic surface θ = const . The cyclotronradius grows in proportion to energies of a particle, γ .Therefore, acceleration is the most effective near the lightsurface, where E θ ≃ B r . As a result, the work of theelectric field ecγB /ω c is just equal to the energy derivedby mc γ particle. Thus, accounting of final cyclotron radiusis very important for acceleration of a charged particle ina magnetosphere especially as it grows with the increase inthe particle energy. The solution of the particle accelerationproblem presented here is based on the integration of theequations of particles motion in an electromagnetic field,which is equivalent to the solution of kinetic equations forindividual components - protons and electrons (we considerthe electron-proton plasma in the magnetosphere ratherthan the electron-positron plasma, which is also possibleunder efficient production of pairs (Beskin, Istomin &Pariev, 1992)).The problem of plasma acceleration by the rotatingneutron star or black hole was also considered in the MHDapproximation in many works, starting from the works ofMichel (1969), (1974). Here we explain in details what theMHD approximation is. In general, a neutral fluid, consistingof positive and negative charges, is described by unifiedequations of hydrodynamics type, in which the interactionof a fluid with the magnetic field is taken into account. Thissuggests (I) that protons and electrons move together as asingle liquid, i.e. u i − u e << u i = u + m e ( u − u e ) /m i ≃ u .Here, u is the average mass fluid velocity. In addition (II),the individual characteristics of the particles disappear,in particular the cyclotron radii of electrons and protons,which, generally speaking, are very different from each other, r ci >> r ce .This means that MHD equations work in regions whosecharacteristic size L significantly exceeds the cyclotronradius of protons, L >> r ci . In MHD equations, whichare hydrodynamics type equations, the ion cyclotron radiusplays the role of the mean free path l in ordinaryhydrodynamics. Therefore, even in weakly collisionalsystems, l ≃ L , MHD is true, if L >> r ci . Underideal conditions, i.e. neglecting dissipation, the MHDapproximation uses the condition of freezing of magneticfield into a liquid, E = − ( u × B ) /c (1). It follows fromOhm’s law j = σ c [ E +( u × B ) /c ] under the condition of highconductivity, σ c → ∞ . However, the very Ohm’s law, calledthe generalized Ohm’s law (Gurnett & Bhattacharjee, 2004),obtained after protons and electrons having been considered,is as follows: j = σ c (cid:26)(cid:20) E + 1 c [ uB ] (cid:21) + 1 enc [ jB ] + 1 enc ∇ P e (cid:27) . (21)Here the quantity P e is the electron pressure. Since j = en ( u i − u e ) , the second term on the right-hand sideof the generalized Ohm’s law is small compared to the firstone if the condition u i − u e << u ≃ u i . In reality, inour problem of centrifugal particle acceleration in the blackhole magnetosphere ( r < r L / sin θ ), the polar electric field E θ (3) does work on both electrons and protons, u iθ < (Fig. 3), u eθ > . Therefore, | ( u i − u e ) θ | = | ( u iθ − | u eθ | ) | > | u iθ | ≃ | u θ | . Thus, we see that the frozen-in condition (1),which is one of the basic postulates of ideal MHD, is notfulfilled in our problem. It is obvious that if E = − u × B /c , then E = − u i × B /c with the accuracy of the small ratio m e /m i << , and the electric field does not work onprotons, u i E = 0 . In terms of physical interpreting the idealMHD does not work here. The ideality implies not only theabsence of Joule heating, but also absence of direct transferof energy from the electric field to charged particles (theliquid is neutral). The acceleration of the flow in MHD canbe only accelerated by two means: the transfer of thermalenergy of the matter into kinetic energy, which is carriedout in hydrodynamics, as well as the transformation of theenergy of the magnetic field into the energy of the flow.This can be seen from the Euler equation in the MHDapproximation: n d p dt = −∇ P − π ∇ B π ( B ∇ ) B . (22)Here, the acceleration occurs due to the pumping ofthe toroidal magnetic field by the central object rotation.Indeed, the radial field rapidly decreases with distance, ∝ r − , but the toroidal field generation, B φ ∝ r − ,takes place due to the freezing-in of the magnetic fieldunder differential rotation of the fluid. With the relation E θ = − r Ω F sin θB r /c = ( u r B φ − u φ B r ) /c , which is the θ component of the equation (1), arising toroidal field is B φ = B r u φ − Ω F r sin θu r . (23)Calculations in the MHD approximation show that theflow acceleration occurs at large distances from the centre, r >> r L , and up to the Lorentz factor γ m = σ / (Michel(1969), (1974); (Beskin, Kuznetsova & Rafikov, (1998)).Here the value of σ is the so-called Michel’s parameter, σ = e ΦΩ F /mc . The flux of the radial magnetic field Φ is, Φ = πB r L . It is easy to see that σ = πω c / Ω F >> .In fact, this is reciprocal to the magnetization parameter, σ ≃ κ − (8). It should be noted that MHD acceleration isalmost isotropic, u ρ ≃ u φ , while the centrifugal accelerationinside the magnetosphere ( r < r L ) discussed in this article,is strongly anisotropic, p φ >> p ρ , p θ . Thus, we see thatcentrifugal acceleration, which is not described in the MHDapproximation, is more effective for the case of a radialmagnetic field (split monopole), γ m ≃ σ / , σ / >>σ / . However this is valid for the radial configurationof the magnetic field. In case are magnetic surfaces ofparabolic configuration, MHD acceleration there can reachto maximum values, γ m ≃ σ (Beskin & Nokhrina, (2006);Tchekovskoy, McKinney & Narayan, (2009)). Here theacceleration is affected by the flow shape, the way it occursin hydrodynamics (nozzle shape). However, in this caseMHD calculations should be performed starting from thelight surface with the boundary conditions specified by theinternal flow, r < r L . ACKNOWLEDGMENTS
This work was supported by Russian Foundation forFundamental Research, grant number 17-02-00788.
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