Quasi-linear diffusion driving the synchrotron emission in active galactic nuclei
aa r X i v : . [ a s t r o - ph . H E ] A p r Astronomy&Astrophysicsmanuscript no. 13515 c (cid:13)
ESO 2018November 16, 2018
Quasi-linear diffusion driving the synchrotron emission in activegalactic nuclei
Z. Osmanov ⋆ and G. Machabeli Center for Theoretical Astrophysics, ITP, Ilia State University, Kazbegi str. 2 a , Tbilisi, 0160, GeorgiaPreprint online version: November 16, 2018 ABSTRACT
Aims.
We study the role of the quasi-linear di ff usion (QLD) in producing X -ray emission by means of ultra-relativistic electrons inAGN magnetospheric flows. Methods.
We examined two regions: (a) an area close to the black hole and (b) the outer magnetosphere. The synchrotron emissionhas been studied for ultra-relativistic electrons and was shown that the QLD generates the soft and hard X -rays, close to the black holeand on the light cylinder scales respectively. Results.
By considering the cyclotron instability, we show that despite the short synchrotron cooling timescales, the cyclotron modesexcite transverse and longitudinal-transversal waves. On the other hand, it is demonstrated that the synchrotron reaction force and aforce responsible for the conservation of the adiabatic invariant tend to decrease the pitch angles, whereas the di ff usion, that pushesback on electrons by means of the aforementioned waves, tends to increase the pitch angles. By examining the quasi-stationary state,we investigate a regime in which these two processes are balanced and a non-vanishing value of pitch angles is created. Key words.
Galaxies:active-Instabilities- Magnetohydrodynamics (MHD)-Radiation mechanisms: non-thermal
1. Introduction
One of the major problems related to active galactic nuclei isthe origin of the nonthermal high energy radiation. Accordingto standard approaches, the most commonly encountered radia-tion mechanisms at a level su ffi cient for application to AGN isthe synchrotron mechanism and the inverse Compton scattering(Blandford et al. 1990). Because of strong synchrotron losses,relativistic electrons in general, quickly lose their perpendicularenergy, and on a synchrotron cooling timescale ∼ − s − − s ,the particles transit to their ground Landau state. In this case,the electrons may be described approximately as moving one-dimensionally along the field lines and the synchrotron radiationto have been absorbed. This is why the broadband emission spec-trum of AGN consists of two components: the high-energy (fromX-rays to γ -rays) component is formed by the inverse Comptonscattering and not by the synchrotron mechanism, which is sup-posed to be involved only in the low-energy (from radio to op-tical / UV) band. However, under certain conditions, due to theQLD of cyclotron waves, the pitch angles might increase, lead-ing to the e ffi cient production of synchrotron radiation.The QLD was applied to pulsars in a series of pa-pers (Machabeli & Usov 1979, Lominadze et al. 1979 ,Malov & Machabeli 2001). Malov & Machabeli (2001)studied optical synchrotron emission of radio pulsars. In theouter parts of pulsar magnetospheres, these authors demon-strated that because of the cyclotron instability, the transversemomenta of relativistic particles is non-zero, giving rise to thepitch angle distribution, which in turn, via the QLD, leads to thesynchrotron emission. Applying the kinetic approach to a partic-ular pulsar, RX J1856.5-3754, Chkheidze & Machabeli (2007) Send o ff print requests to : Z. Osmanov ⋆ E-mail: [email protected] (ZO); [email protected](GM) showed that waves excited by the cyclotron mechanism, interms of the creation of the pitch angles, come into the radiodomain. The QLD interesting because the recent detectionof very high energy (VHE) pulsed emission form the Crabpulsar (Albert et al. 2008). The MAGIC Cherenkov telescopediscovered the pulsed emission above 25GeV between 2007October and 2008 February. It has been shown that the corre-sponding VHE signal peaks at the same phase as the signal inthe optical spectrum (Albert et al. 2008). In turn this indicatesthat the polar cap models must be excluded from the possiblescenario of the radiation. On the other hand, analysis of theMAGIC data implies that the location of the aforementionedVHE and optical radiation must be the same. According tothe quasi-linear di ff usion, on length scales typical of the lightcylinder (a hypothetical zone, where the linear velocity ofrigid rotation equals exactly the speed of light), the cyclotroninstability occurs in the optical band, leading to an increase inthe pitch angles via the QLD. This mechanism automaticallyexplains the coincidence of phases in the optical and VHE bands(Machabeli & Osmanov 2009 ).AGN magnetospheres are supported by strong magneticfields and therefore, the QLD might also be of great impor-tance to these particular objects. As aforementioned, for ultra-relativistic electrons the synchrotron losses are so e ffi cient thatthe synchrotron mechanism takes place only for relatively lowenergy particles and highly relativistic electrons are involved inradiation via the inverse Compton scattering. This is not true forthe QLD, because as for the pulsar magnetospheres, AGN mag-netospheric particles will undergo the QLD, preventing the rapiddamping of pitch angles, giving rise to the emission process.In the present paper, we study the role of the QLD in produc-ing the X-rays via the synchrotron mechanism in AGN magneto-spheres. The paper is organized as follows. In Section 2, we con- Z. Osmanov and G. Machabeli: Quasi-linear di ff usion driving the synchrotron emission in active galactic nuclei sider the kinetic approach to the quasi-linear di ff usion, in Sect.3 we present our results and in Sect. 4 we summarize them.
2. Main consideration
When relativistic particles move in the magnetic field, they emitelectromagnetic waves corresponding to the photon energies(Rybicki & Lightman 1979) ǫ keV ≈ . × − B γ sin ψ, (1)where by B we denote the magnetic induction, γ is the Lorentzfactor of particles, and ψ denotes the pitch angle. Equation (1)represents the photon energy in the maximum emission inten-sity. As we already mentioned in the introduction, the timescaleis very short for the transit to the ground Landau state thatprovides quasi-one-dimensional motion of electrons along thefield lines without radiation. As investigated in a series of pa-pers (Machabeli & Usov 1979, Malov & Machabeli 2001), thecyclotron instability of the electron-positron plasma under cer-tain conditions, may however ”create” pitch angles, which acti-vate the subsequent synchrotron process.We consider the plasma to consist of two components: (a)the electron-positron plasma component with the Lorentz fac-tor, γ p and (b) highly relativistic electrons, the so-called beamcomponent with the Lorentz factor, γ b ( γ b ≫ γ p ). According tothe the QLD model, the consequent transverse modes generatefrequencies (Kazbegi et al. 1992) ω t ≈ kc (1 − δ ) , (2)where δ = ω p ω B γ p . (3)We denote by k the modulus of the wave vector, where c is thespeed of light, ω p ≡ p π n p e / m is the plasma frequency, ω B ≡ eB / mc is the cyclotron frequency, e and m are the electron chargeand the rest mass, respectively, and n p is the plasma density.Kazbegi et al. (1992) demonstrated that the aforementionedwaves are excited if the cyclotron resonance condition ω − k k V k − k x u x ± ω B γ b = , (4)is satisfied, where u x ≡ cV k γ b /ρω B denotes the drift velocityof particles, k k is the wave vector’s longitudinal (parallel to thebackground magnetic field) component, k x is the wave vector’scomponent along the drift, V k is the component of velocity alongthe magnetic field lines, and ρ is field line’s curvature radius. Bytaking into account the resonance condition, from Eq. (2), onecan obtain an expression for the excited cyclotron frequency ω ≈ ω B δγ b . (5)In deriving Eq. (5), we have taken into account the condition λ > n − / p , which means that in exciting waves all resonanceparticles participate (collective phenomena), therefore the rangeof spectral frequencies is wide. In general, the cyclotron modeexcites if the distribution function, f , is almost one dimensionaland depends on the longitudinal momentum. On the other hand,the magnetic field in AGN magnetospheres is strong enough tomaintain the frozen-in condition, which in turn means that theparticles follow field lines and thus, f behaves according to p || . When particles move in a nonuniform magnetic field, theyundergo a force G that is responsible for the conservation ofthe adiabatic invariant, I = cp ⊥ / eB (Landau & Lifshitz 1971).The corresponding components of this force are given by G ⊥ = − mc ρ γ b ψ, G k = mc ρ γ b ψ . (6)In the synchrotron regime, we should detect the radiative force(Landau & Lifshitz 1971): F ⊥ = − αψ (1 + γ b ψ ) , F k = − αγ b ψ , (7)where α = e ω B / (3 c ).These forces ( F , G ) tend to decrease the pitch angle of theparticle. On the other hand, the feedback of low frequency wavesexcited by particles be means of the cyclotron resonance, leadsto the quasi-linear di ff usion of particles. In turn, the QLD, at-tempts to widen the range of the pitch angles opposing both F and G . The dynamical process saturates when the e ff ects of theabove-mentioned forces are balanced by the di ff usion. There are,in general two di ff erent mechanisms of radiation: (I) the reso-nance cyclotron emission and (II) the synchrotron process, thefirst of which, as we have already mentioned, is a collective phe-nomenon ( λ > n − / p ), whereas the second is a single particleprocess ( λ < n − / p ) that does not require superposition.We consider the case | G ⊥ | ≫ | F ⊥ | and | G k | ≪ | F k | . By assum-ing a quasi-stationary scenario ( ∂/∂ t = mc γ b ψ ∂∂ψ (cid:0) ψ G ⊥ f (cid:1) + mc ∂∂γ b (cid:16) F k f (cid:17) + υ ∂ f ∂ r == m c γ b ψ ∂∂ψ ψ D ⊥⊥ ∂ f ∂ψ ! + mc ψ ∂∂ψ ψ D ⊥k ∂ f ∂γ b ! ++ mc ∂∂γ b ψ D ⊥k ∂ f ∂ψ ! , (8)where f = f ( ψ, p k ) is the distribution function of particles, p k isthe longitudinal momentum, D ⊥⊥ ≈ π e c ω p ω B γ b | E k | , D ⊥ k ≈ − π e cp k | E k | , (9)are the di ff usion coe ffi cients, n b is the density of the beam com-ponent, and | E k | is the energy density per unit wavelength. Thecorresponding energy density can be estimated to be | E k | k . Weassume that half of the plasma energy density, mc n b γ b / | E k | is given by | E k | = mc n b γ b ω . (10)By expressing the distribution function as χ ( ψ ) f ( p k ), one cansolve Eq. (9) for χχ ( ψ ) = C e − A ψ , (11)where A ≡ m c γ b (cid:16) ω B /ω p (cid:17) π e ρ | E k | γ p , (12) . Osmanov and G. Machabeli: Quasi-linear di ff usion driving the synchrotron emission in active galactic nuclei 3 l og ε k e V ω (MHz) Fig. 1.
The synchrotron emission energy versus the cyclotronfrequency. The set of parameters is M BH = M ⊙ , n b = cm − , B = G , γ p = γ b ∈ { × ; 10 } , and ρ = R g .As we see from the plot, the low frequency cyclotron mechanismarises in the radio band, inducing the QLD and subsequently cre-ating relatively high pitch angles, which in turn lead to the X-rayemission.The corresponding mean value of the pitch angle can be es-timated as to be¯ ψ = R ∞ ψχ ( ψ ) d ψ R ∞ χ ( ψ ) d ψ = √ π A . (13)Therefore, as we see, the QLD maintains the pitch angles andprevents them from damping, which in turn maintains the prop-erties of the synchrotron process.
3. Discussion
Most AGN exhibit VHE emission, which in turn indi-cates that the AGN magnetosphere is contained of ultra-relativistic electrons. In this context, the origin of accelerationof particles is very important. Proposed mechanisms such asFermi-type acceleration (Catanese & Weeks 1999), centrifugalacceleration (Machabeli & Rogava 1994, Osmanov et al. 2007,Rieger & Aharonian 2008), and acceleration due to the blackhole dynamo mechanism (Levinson 2000) can e ff ectively pro-vide Lorentz factors of the order of ∼ − .For an isotropic distribution of relativistic electrons(Blandford et al. 1990) in very strong magnetic fields, parti-cles emit in the synchrotron regime with the power, P ≈ e B β ⊥ γ / (3 m c ), therefore by assuming β ⊥ ∼
1, the syn-chrotron cooling timescale, t s ≡ γ mc / P can be estimated tobe t s ≈ × − × GB ! × γ ! s , (14)where we have taken into account that the energy is uni-formly distributed between the beam and the plasma compo-nents, n b γ b ≈ n p γ p , and the magnetic field is normalized to thetypical value of the magnetic induction close to the supermassiveblack hole (Thorne et al. 1988). As is clear from this expression,the synchrotron timescale for ultra-relativistic electrons in the
35 40 45 50 55 601.822.22.42.62.83 ε k e V ω (kHz) Fig. 2.
The synchrotron emission energy versus the cyclotronfrequency. The set of parameters is n b = cm − , L = erg / s , Ω = × − s − , γ p = γ b ∈ {
4; 5 }× and ρ = r lc .As is clear, the low frequency cyclotron mechanism arises in theradio band, inducing via the QLD, the X-ray emission.very vicinity of the black hole is so short ( ∼ − s ) that theseparticles very rapidly lose their perpendicular kinetic energy andtransit to the ground Landau state. One can obtain a similar resultfor outer magnetospheric lengthscales, thus for the light cylinderarea, when the magnetic field is determined by the bolometricluminosity, L of AGN to be the equipartition field B lc ≈ s Lr lc c ≈ × L erg / s ! / × Ω × − s − ! G , (15)where r lc = c / Ω is the light cylinder radius and Ω the angular ve-locity of rotation. If we consider a typical value of the bolomet-ric luminosity, ∼ erg / s , then one can see from Eq. (15) thatthe magnetic induction equals approximately 290 G , which, com-bined with Eq. (14), leads to the cooling timescale, 5 . × − s .All quantities vary because of the cyclotron instability, whichcauses to the QLD. By using Eqs. (3-5), we can estimate thefrequency of the cyclotron mode to be ω ≈ . × (cid:18) γ p (cid:19) γ b ! (cid:18) B G (cid:19) × cm − n b ! Hz , (16)We consider a nearby zone of the AGN with mass M BH = M ⊙ ( M ⊙ is the solar mass) and typical magnetospheric parameters n b = cm − , B = G and γ p =
2. Then, examining theultra-relativistic beam component electrons with Lorentz factors γ b ∈ {
3; 4 } × and assuming that the curvature radius ρ of fieldlines is of the order of the gravitational radius, R g ≡ GM BH / c ,one can see from Eq. (13) that the created pitch angle is of the or-der of 10 − rad , which indicates the synchrotron emission in the X -ray domain [see Eq. (1)]. In Fig. 1, we show the synchrotronemission energy versus the excited cyclotron frequency. As isclear from the figure, the cyclotron instability appears in the ra-dio band leading to the X -ray emission. One can see that theradio emission is generated by the collective phenomena. Weconsider the radio frequency 5 MeV , corresponding to the wave-length, λ , of the order of 6000 cm . On the other hand, the av-erage distance between particles, d = n − / p is of the order of Z. Osmanov and G. Machabeli: Quasi-linear di ff usion driving the synchrotron emission in active galactic nuclei
45 45.2 45.4 45.6 45.8 46 46.2 46.4 46.6 46.8 4700.511.522.53 ε k e V log(L) Fig. 3.
The synchrotron emission energy versus the AGN lu-minosity. The set of parameters is n b = cm − , L = − erg / s , Ω = × − s − , γ p = γ b = × and ρ = r lc .2 × − cm , which is shorter by many orders of magnitude than λ .This in turn, indicates that emission in the radio band is providedby the collective phenomena. In contrast to this, the VHE radi-ation generated in the X -ray domain (see Fig. 1) is a single par-ticle mechanism. The importance of the quasi-linear di ff usion istwofold: (a) it generates the synchrotron radiation in strong mag-netic fields for ultra-relativistic particles, which would be impos-sible without the QLD and (b) it simultaneously excites emis-sion in two di ff erent domains. In this context, it is easy to checkwhether the emission is driven by the QLD by verifying that(I) the radiation is linearly polarized and (II) both signals haveequal phases. In (Machabeli & Osmanov 2009 ), we performedthe same theoretical analysis of the observed VHE ( > GeV )pulsed emission of the Crab pulsar (Albert et al. 2008).Unlike the previous case, we show in Fig. 2 the same behav-ior for the outer magnetospheric (light cylinder) lengthscales.The set of parameters is M BH = M ⊙ , n b = cm − , L = erg / s , Ω = × − s − , γ p = γ b ∈ {
3; 4 } × and ρ = r lc . As shown in Fig. 2, the radio frequency close to the lightcylinder zone, in the kHz domain excites the hard X -ray emis-sion by means of the QLD. From Eqs. (6,7), one can straight-forwardly check the validity of our assumptions, | G ⊥ | ≫ | F ⊥ | and | G k | ≪ | F k | , confirming our approach. It is also interest-ing to investigate the QLD for di ff erent values of the luminos-ity. We examine the luminosity interval from 10 erg / s to theEddington limit, which for the given black hole mass, 10 M ⊙ ,equals 10 erg / s . In Fig. 3, we show the strong X -ray energyversus the AGN luminosity. The set of parameters is M BH = M ⊙ , n b = cm − , L = − erg / s , Ω = × − s − , γ p = γ b = × and ρ = r lc . As is evident from the figure,the emission energy is a continuously decreasing function of theluminosity. Indeed, by combining Eqs. (1,12,13,15), we see thatthe synchrotron emission energy behaves as L − / .Therefore, as our investigation shows, the QLD is a workingmechanism in AGN magnetospheric flows and drives the syn-chrotron process.
4. Summary
The main aspects of the present work can be summarized as fol-lows:1. We studied the quasi-linear interaction of proper modes ofAGN magnetospheric plasmas with the resonant plasma par-ticles. For this purpose, the synchrotron reaction force hasbeen taken into account. The role of the QLD was studiedin the context of producing the soft and hard X -ray emissionfrom AGN.2. It has been shown that the synchrotron cooling timescalesfor ultra-relativistic electrons are very small, and particlesrapidly transit to the ground Landau state, that in turn, pre-vents the subsequent radiation. We found that, under certainconditions the cyclotron instability develops, leading to thecreation of pitch angles, and the subsequent synchrotron pro-cess.3. We have considered two extreme regions of magnetospheres:(a) relatively close to the black hole and (b) the light cylin-der zone. As our model shows, the cyclotron instability, un-der certain conditions, generates the radio frequency in therange (0 . − MHz and creates the soft X -ray emission,(0 . − keV , via the QLD. For the light cylinder area, ra-dio spectra occurs in the range (38 − kHz , which producesthe hard X -ray emission in the domain (1 . − . keV . Wehave emphasized that from an observational evidence onecan directly verify the validity of the QLD by determining(I) the polarization and (II) phases of signals in radio and X -ray domains respectively.4. The quasi-linear di ff usion has also been studied versus theAGN luminosity. It was shown that for more luminous AGN,the corresponding photon energy of the hard X -ray emission,generated by the synchrotron mechanism is lower. Acknowledgments