On the collective curvature radiation
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 11 November 2018 (MN L A TEX style file v2.2)
On the collective curvature radiation
Ya. N. Istomin , , A. A. Philippov and V. S. Beskin , P.N.Lebedev Physical Institute, Leninsky prosp., 53, Moscow, 119991, Russia Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, 141700, Russia
Accepted, Received
ABSTRACT
The paper deals with the one possible mechanism of the pulsar radio emission, i.e., withthe collective curvature radiation of the relativistic particle stream moving along thecurved magnetospheric magnetic field lines. It is shown that the electromagnetic wavecontaining one cylindrical harmonic exp { isφ } can not be radiated by the curvatureradiation mechanism, that corresponds to radiation of a charged particle moving alongcurved magnetic field lines. The point is that the particle in vacuum radiates the triplexof harmonics ( s, s ± ε ( ω, k , r ), that contains the information on the reaction on allpossible types of radiation. Key words:
Radio pulsars
The curvature radiation is the type of the bremsstrahlungradiation when a radiated charged particle moves along thecurved trajectory with the curvature radius ρ and its accel-eration is orthogonal to the velocity v . The cyclotron rota-tion of a charged particle in the external magnetic field B isthe example of this motion when ρ = v ⊥ /ω c . Here ω c is thecyclotron frequency, ω c = eB/m e cγ , e and m e are the chargeand the mass of a particle, γ is the particle Lorentz factor,and v ⊥ is the component of the particle velocity which isorthogonal to the magnetic field.Moving along the circular trajectory, a particle radiatesat harmonics of the cyclotron frequency: ω = nω c . This ra-diation is called as cyclotron radiation for a nonrelativisticparticle and as synchrotron radiation for a relativistic parti-cle ( γ (cid:29) n (cid:39) γ . The total radiation power I also growswith the particle energy, I ∝ γ . Therefore, the synchrotronradiation of relativistic particles is presented widely in thespace radiation (Ginzburg & Syrovatskii 1964).It is necessary to stress that the length of formationof the curvature radiation, though is larger than the wavelength λ , is much less than curvature radius ρ . So, the prop-erties of the curvature radiation do not differ from that ofthe synchrotron radiation in which the cyclotron radius isequal to the local curvature radius ρ . The frequency of themaximum of the spectral power, ω (cid:39) cγ /ρ , and radia-tion power, I = 4 / e cγ /ρ , increases with the particle energy. Here the dependence on γ is stronger than for thesynchrotron radiation because the curvature is fixed, anddoes not fall with the energy as for the motion in the con-stant magnetic field.The curvature mechanism of radiation is believed to beconnected with the mechanism of the coherent pulsar ra-dio emission. Indeed, in the region of the open magneticfield lines in the pulsar magnetosphere there is a relativisticelectron-positron plasma moving with relativistic velocitiesalong curved magnetic field lines. For the typical values ofcurvature radius, ρ (cid:39) cm, and Lorentz factor of elec-trons and positrons, γ (cid:39) , the characteristic frequencyof the curvature radiation is in the radio band. In the mag-netosphere, where the curvature frequency coincides withthe plasma frequency, ω p /γ / , we can expect the collectivecurvature radiation. Here ω p = (4 πe n e /m e ) / is the usualplasma frequency, and n e is the plasma number density. Inthe strong magnetic field of the pulsar magnetosphere, whenthe charged particles can move along magnetic field linesonly, the frequency of the plasma oscillations is γ − / timesless than the usual plasma frequency ω p .It seems natural to continue the analogy between thecurvature radiation and the cyclotron radiation for the col-lective radiation. But there is the essential difference be-tween them, which does not permit to rewrite formulas of cy-clotron plasma radiation for the curvature radiation replac-ing the cyclotron radius by the curvature radius. The matteris that at each point of plasma in the magnetic field the dis-tribution of particles over transverse velocities is isotropic.All directions of particle transverse motion exist, so that the © a r X i v : . [ a s t r o - ph . H E ] N ov Ya. N. Istomin, A. A. Philippov, and V. S. Beskin average velocity equals zero. It is not so for the curvature ra-diation when all particles have only one direction of motionalong the magnetic field.For the plasma physics the problem of the collectivecurvature radiation is rather complicated since it demandsthe consideration of an essentially nonuniform plasma. Itdoes not result from the change of parameters of magneticfield and plasma in space. These effects can be taken intoaccount in the local approximation because the wave lengthof radiation is much less than the scales of inhomogeneities.In order not to lose the curvature radiation we need to in-clude into consideration the turn of the vector of anisotropyof the particle distribution function, f ( p ) ∝ δ ( p − p (cid:107) B /B ),in space. Here p (cid:107) is the longitudinal particle momentum.Two parameters of the curvature radiation, i.e., thelength of formation l f = ρ /γ (cid:39) λγ (cid:29) λ and the width ofthe radiation directivity δφ (cid:39) γ − , connect by the relation l f /ρ (cid:39) δφ . Thus, the particle is in the synchronism with thewave (i.e., the particle sees the constant wave phase) alongthe path on which the wave intensity changes essentially.The value of λ is the wave length of the curvature radiation, λ (cid:39) ρ /γ .The problem of calculation of the dielectric permittivityin the geometrical optics approximation for the nonuniformanisotropic plasma particle distribution was solved by Be-skin, Gurevich and Istomin (below BGI, 1993). They alsodescribed the collective curvature-plasma interaction, whenthe electromagnetic waves, connected with the curvatureradiation, are amplified simultaneously by the Cherenkovmechanism. This effect is absent in the vacuum. However,this procedure is rather complicated and demands clear un-derstanding. Because of that there are some incorrect state-ments in the literature (see, e.g., Nambu 1989; Machabeli1991, 1995).Apart, another way of investigation of the problem ofthe collective curvature radiation was carried out duringmany years (Asseo et al. 1983; Larroche & Pellat 1987; Lyu-tikov et al. 1999; Kaganovich & Lyubarsky 2010). They con-sidered more simple task connected with the pure cylindricalgeometry which can be solved ”exactly”. In such a statementthe magnetic field lines are considered to be concentric, therelativistic plasma moving (i.e., rotating) along the mag-netic field lines owing to the centrifugal drift directed par-allel to the cylindrical axis ( z -coordinate) with the velocity u = cρ c /ρ (cid:28) c . Here again ρ c = c/ω c . But this approachcannot be used when analyse the curvature radiation (Be-skin, Gurevich & Istomin 1988).Indeed, let us choose the electromagnetic fields of thewave, as was done in all the papers mentioned above, in theform( E , B ) = ( E ( ρ ) , B ( ρ )) × exp {− iωt + isφ + ik z z } . (1)Here ω is the wave frequency, s is integer number definingthe azimuthal wave vector k φ , and k z is the longitudinalwave vector along the cylinder. In this approach the waveamplitudes E ( ρ ) , B ( ρ ) are to be considered as functions ofthe radial distance ρ only. Moreover, not vectors E and B , but their cylindrical components ( E, B ) ρ , ( E, B ) φ and( E, B ) z depend on the coordinate ρ only. It means that thewave polarization follows the magnetic field, turning fromone point φ to another. It can be so if we have the defi-nite boundary condition, e.g., putting the system into the metallic coat. Under such suggestions we come to the onedimensional problem, which can be easily solved. Here wewill show that such a wave does not have any relation to thecurvature radiation.Really, let us consider the particle moving exactly alongthe circle of radius ρ with the constant velocity v ; this mo-tion corresponds to the infinite magnetic field. Then the ra-diated power is equal to the work of the wave electric fieldunder the particle electric current. The electric current is j = evδ ( φ − Ω t ) δ ( z ) δ ( ρ − ρ ) ρ e φ , (2)where Ω = v/ρ , and for selected polarization we get (cid:90) jE d r = evE φ ( ρ ) exp {− iωt + is Ω t } . (3)As we see, the radiation is possible only if ω − s Ω = 0, i.e., ω = k φ v . It is just the condition of Cherenkov, not curvatureradiation. The point is that the wave with such polarizationcan not be radiated by the curvature mechanism. The differ-ence between the curvature wave and the Cherenkov waveis in the finite interaction time of the bremsstrahlung radi-ation with a radiated particle. The freely propagating wavewith almost constant polarization deflects from the direc-tion of a particle motion. As a result, the nonzero projec-tion of the wave electric field on the particle velocity (i.e.,on the direction of the electric current) occurs, and the wavetakes away the energy from the particle. This continues thefinite time τ = l f /v that can be determined from the re-lation τ ( ω − kv ) (cid:39)
1. For the relativistic particle ( v (cid:39) c ) τ = ( ρ /ωc ) / (cid:39) ρ /cγ . Below we will find the real polar-ization of the curvature radiation.The paper is organized as follows. In section 2 we willfind that the polarization of the curvature wave does notcorrespond to one cylindrical harmonic. In section 3 it isshown that the nonlinear wave interaction can lead to sig-nificant changes in cylindrical modes propagation. In section4 the BGI permittivity tensor will be derived from the per-mittivity corresponding to one cylindrical mode. Finally, insection 5 we discuss the main results of our consideration. The radiation field of the electric current density j and theelectric charge density ρ e of the moving particle with thecharge e is described by the retarded potentials (Landau &Lifshits 1975): A = 1 c (cid:90) j ( t (cid:48) ) R d r , (4)Φ = (cid:90) ρ e ( t (cid:48) ) R d r . (5)Here t (cid:48) = t − R/c is the retarded time, R is the distance fromthe charge location at the time t (cid:48) to the observer which hasthe cylindrical coordinates ( ρ, φ, z ), R = (cid:2) ρ + z + ρ − ρρ cos( φ (cid:48) − φ ) (cid:3) / , (6) φ (cid:48) = Ω t (cid:48) . (7) © , 000–000 n the collective curvature radiation After the Fourier transformation of potentials (4)–(5) overthe time we obtain A ω = 12 π (cid:90) A ( t ) exp { iωt } d t, (8)Φ ω = 12 π (cid:90) Φ( t ) exp { iωt } d t. (9)It is convenient now to replace the integration over time t bythe integration over the retarded time t (cid:48) and then over theangle ( φ (cid:48) − φ ). As a result one can obtain for the cartesiancomponents ( x, y, z ) of the vector potential A and the scalarpotential Φ[ A ω ; Φ ω ] = eρ πc exp { iωφ/ Ω } (cid:104) − K s ; K c ; 0; cv K (cid:105) . (10)Here the quantities K , K s , K c are the functions of coordi-nates ρ and z only and they are equal to K = (cid:90) exp { iω ( R/c + Ω − ) α } R + vρ sin α/c d α,K s = (cid:90) exp { iω ( R/c + Ω − ) α } sin αR + vρ sin α/c d α, (11) K c = (cid:90) exp { iω ( R/c + Ω − ) α } cos αR + vρ sin α/c d α,R = ( ρ + z + ρ − ρρ cos α ) / . The expression (10) is valid at any point r , i.e., not only inthe wave zone. The dependence over the angle φ is given bythe exponent exp { iωφ/ Ω } . From the periodicity over φ wehave ω = s Ω.The key point of the above expansion (10) is that theradiated wave is the superposition of three harmonics: s , s − s + 1. For example, the azimuthal electric field E φω isequal to E φω = iωv (cid:18) − ρ ρ Φ ω + vc A φω (cid:19) = − i eρ ω πv e isφ (cid:20) ρ ρ K − v c ( K s sin φ + K c cos φ ) (cid:21) . (12)The first term in Eqn. (12), which is proportional to thescalar potential Φ, is not important in the wave zone, ρ (cid:29) ρ , but is significant in the near zone on the parti-cle trajectory ρ = ρ . Due to this term, the particle, whichis in the resonance with one of three harmonics, say with s ( ω = s Ω), is beaten out of the synchronism by neigh-bour harmonics s ±
1. The electric field E φω changes itssign during the time τ . The synchronism condition, i.e.,1 − cos Ω τ (cid:39) − v /c = γ − , defines the time τ , τ (cid:39) / Ω γ = ρ /cγ, (13)which coincides with the time of formation of the curvatureradiation.Thus, the radiated curvature wave consists of three har-monics s, s ± s ± s = ω/ Ω is due to the additionalmodulation of the radiation field induced by a modulationof the particle electric current having the harmonic s = 1.Now one can understand why the simple problem of the col-lective curvature radiation in the cylindrical geometry withonly one azimuthal harmonic exp { isφ } does not reveal any significant amplification of waves (Asseo et al., 1983; Lyu-tikov et al., 1999; Kaganovich & Lyubarsky, 2010). In thiscase the chosen wave polarization does not contain primor-dially the curvature mechanism. In the previous section it was shown that the curvature radi-ation of one charged particle can not be described in the purecylindrical geometry by one azimuthal harmonic exp { isφ } .In a collective radiation the modulation of the particle elec-tric current appears together with electromagnetic field ex-citation. Because of that the resonant azimuthal harmonic s = ωρ/v φ mixes with harmonics of the electric current mod-ulation and produces all possible values of s . Further in thesection 4 we will see the all azimuthal harmonics s give con-tribution to the response of a media on an electromagneticfield. But in this section it will be demonstrated that thecollective curvature radiation of only triplex of azimuthalharmonics ( s, s ±
1) differs significantly from that of oneharmonic s as it is usually considered in the literature.Let us consider the simple cylindrical one-dimensionalproblem of radiation of the cold stream of plasma particleswith the charge e and the mass m e moving along the infiniteazimuthal magnetic field B = B φ . In this case the particlescan move only in φ -direction with the velocity v φ at differentcylindrical radius ρ . The unperturbed particle density n (0) and velocity v (0) φ are constants, i.e., they do not depend on ρ . The electric current j has only φ -component as well as B z -component of the wave magnetic field ( B ρ = B φ = 0).Accordingly, the wave electric field has two components E ρ and E φ ( E z = 0).The dependence of the wave fields over time and coor-dinates is the following[ E ρ ; E φ ; B z ] = [ E ρ ( ρ ); E φ ( ρ ); B z ( ρ )] exp {− iωt + isφ } . (14)Then, we obtain from Maxwell equationsd E ( σ ) φ d ρ = iσρ E ( σ ) ρ − i ρσ ω c E ( σ ) ρ − E ( σ ) φ ρ , (15)d E ( σ ) ρ d ρ = − i σρ E ( σ ) φ + 4 πω σρ j ( σ ) φ − E ( σ ) ρ ρ . (16)Here index σ corresponds to one of three harmonics s or s ±
1. For simplicity we use here the dimensionless variable r defined as r = ρω/c , as well as quantities Λ = ω p / ( ω γ )and J σ = 4 πj ( σ ) φ / (Λ ω ). Here ω p = (4 πne /m e ) / is plasmafrequency, and γ is the Lorentz-factor of the particle motion: γ = (1 − v φ /c ) − / . After these definitions the equationsabove take the following formd E ( σ ) φ d r = iσr E ( σ ) ρ − i rσ E ( σ ) ρ − E ( σ ) φ r , (17)d E ( σ ) ρ d r = − i σr E ( σ ) φ + Λ σr J σ − E ( σ ) ρ r . (18)As was already stressed, we consider here the interac-tion of three waves s, s ±
1. It is important that they arenot independent and their interaction is realized by thestatic electric field [ E ρ ( ρ ); E φ ( ρ )] exp { iφ } having the firstazimuthal harmonic s = 1. This electrostatic field turnsto be the result of nonlinear interactions of high frequency © , 000–000 Ya. N. Istomin, A. A. Philippov, and V. S. Beskin neighbour harmonics s and s ±
1. Equations for the mode s = 1 under the same definitions ared E φ d r = ir E ρ − E φ r , (19)d E ρ d r = − i r E φ + Λ Z − E ρ r . (20)Here Z = 4 πnec/ (Λ ω ).To determine the response of the stream on the electro-magnetic fields of the wave one can use the continuity andEuler equations ∂n∂t + ∇ ( n v ) = 0 , (21) (cid:18) ∂∂t + v ∇ (cid:19) p = e (cid:16) E + (cid:104) v c , B (cid:105)(cid:17) . (22)It is easily to understand that only the φ -component ofEuler equation is needed, while the radial component justprovides us the equilibrium configuration across the infinitemagnetic field. We represent the plasma number density and the plasma velocity as the expansion over powers of the waveamplitude v φ = v (0) φ + δv (1) φ + δv (2) φ + ..., (23) n = n (0) + δn (1) + δn (2) + .... (24)The linear response can be easy found n (1) = n (0) kv (1) φ ω − kv (0) φ , (25) v (1) φ = i eE φ m e γ ( ω − kv (0) φ ) , (26)where k = s/ρ . On the other hand, for the nonlinear currentthe nonlinear relation between δv φ and δp φ should be takeninto account δp φ = m e γ δv φ − m e v (0) φ γ ( δv φ ) c . (27)The result of cumbersome but straightforward calculation is J s = 11 − sv (0) φ /r (cid:34) i E sφ − sv (0) φ /r + α rv (0) φ (cid:32) A s,s − E s − φ E φ − ( s − v (0) φ /r − A s,s +1 E s +1 φ E ∗ φ − ( s + 1) v (0) φ /r (cid:33)(cid:35) , (28) J s − = 11 − ( s − v (0) φ /r (cid:34) i E s − φ − ( s − v (0) φ /r − α rv (0) φ A s,s − E sφ E ∗ φ − sv (0) φ /r (cid:35) , (29) J s +1 = 11 − ( s + 1) v (0) φ /r (cid:34) i E sφ − ( s + 1) v (0) φ /r + α rv (0) φ A s,s +1 E sφ E φ − sv (0) φ /r (cid:35) , (30) Z = 1 (cid:16) v (0) φ (cid:17) (cid:34) i E /r + α (cid:32) E s +1 φ E s ∗ φ (1 − ( s + 1) v (0) φ /r )(1 − sv (0) φ /r ) + E sφ E ( s − ∗ φ (1 − sv (0) φ /r )(1 − ( s − v (0) φ /r ) (cid:33)(cid:35) , (31) A i,j = 11 − iv (0) φ /r + 11 − jv (0) φ /r − γ , Here α = e/ ( m e cγ ω ) is the particle velocity dividedover the velocity of light. The same quantities for planewaves can be found in (BGI, 1993). Equations above areevaluated with vacuum initial condition for the normal modethat can be presented analytically, E ( σ ) φ = − J (cid:48) σ ( r ) , E ( σ ) r = iσJ σ ( r ) /r . Here J σ ( r ) is the Bessel function. It shouldbe noted that the singularity in equations (17) is passedsmoothly by additional small term + iε in the resonance de-nominators in (25)–(26).In numerical calculations equations (17)–(20) for σ = s and σ = s ± J σ and Z . In the first case we neglect non-linearterms in (28)–(31), while the second one corresponds to thefull non-linear problem. On Fig. 1 the results obtained forthis cases are presented. For better representation of the in-fluence of the nonlinear current, we choose the amplitudesof s − s + 1 modes twenty times higher than the am-plitude of the s mode. In reality the s -mode interacts withthe whole continuum of modes, so this model assumption israther reasonable. Fig. 1 shows that in this case the inten-sity of the wave | E | is approximate 2.5 times larger than in the case when the nonlinear current is neglected. Hence, onecan conclude that three wave interaction is rather effective.Thus, we have shown that the triplex of cylindrical har-monics, which corresponds better to the curvature mech-anism, is amplified more effective than the separated har-monic having the single value of the azimuthal number. Infact the real polarization of the collective curvature modecan be obtained only by calculating the permittivity tensorof the streaming plasma in the strong curved magnetic field.The solution of wave equations produces not only the disper-sive equation for normal waves, ω = ω ( k ), but defines alsotheir polarization. A priory it is unclear what polarizationcorresponds to unstable modes.At first sight, the problem considered above is essen-tially nonlinear and has no direct connection with the ques-tion of the linear wave amplification. We included nonlin-earity only in order to connect harmonics s, s ± s ± s -mode am-plification. It is clear also that interaction of s ± s = 1 will result in all azimuthal harmonics. © , 000–000 n the collective curvature radiation Figure 1.
Model calculations of two cases, Λ = 10 − , ν = 1 GHz , γ = 5, s = 125. In this section we will show that the asymptotic behaviourof the BGI dielectric tensor in the case of large enough cur-vature radius ρ can be found directly from the plasma re-sponse on the one cylindrical mode. For the infinite toroidalmagnetic field only the response to the toroidal componentof the wave electric field E φ is to be included into considera-tion (Beskin 1999). Here and below we consider the station-ary medium only, so the time dependence can be chosen asexp {− iωt } . Making summation over all cylindrical modes,one can write down D φ ( ρ, φ ) = E φ ( ρ, φ ) − ∞ (cid:88) s = −∞ E φ ( ρ, s ) K ( ρ, s ) exp { isφ } , (32)where K ( ρ, s ) = 4 πe ω (cid:90) v φ ω − sv φ /ρ ∂f (0) ∂p φ d p φ . (33)Here f (0) ( p φ ) is unperturbed distribution function. Makingthe Fourier transformation E φ ( ρ, s ) = 12 π π (cid:90) E φ ( ρ, φ (cid:48) ) exp {− isφ (cid:48) } d φ (cid:48) (34) and the transition to cartesian coordinate system one canobtain: D x = E x + 12 π (cid:90) ρ (cid:48) d ρ (cid:48) d φ (cid:48) ρ (cid:48) ∞ (cid:88) s = −∞ E φ ( ρ (cid:48) , φ (cid:48) ) δ ( ρ − ρ (cid:48) ) × K ( ρ, s ) exp { is ( φ − φ (cid:48) ) } sin φ, (35) D y = E y − π (cid:90) ρ (cid:48) d ρ (cid:48) d φ (cid:48) ρ (cid:48) ∞ (cid:88) s = −∞ E φ ( ρ (cid:48) , φ (cid:48) ) δ ( ρ − ρ (cid:48) ) × K ( ρ, s ) exp { is ( φ − φ (cid:48) ) } cos φ. (36)We choose the local coordinate system with the y -axis di-rected along the magnetic field and the x -axis, that is or-thogonal to it. From the above equations one can obtain thepermittivity kernel components ε yy ( r , r (cid:48) ) = 1 − π ρ (cid:48) ∞ (cid:88) s = −∞ δ ( ρ − ρ (cid:48) ) K ( ρ, s ) × exp { is ( φ − φ (cid:48) ) } cos φ cos φ (cid:48) ; (37) ε yx ( r , r (cid:48) ) = 12 π ρ (cid:48) ∞ (cid:88) s = −∞ δ ( ρ − ρ (cid:48) ) K ( ρ, s ) × exp { is ( φ − φ (cid:48) ) } cos φ sin φ (cid:48) ; (38) ε xy ( r , r (cid:48) ) = 12 π ρ (cid:48) ∞ (cid:88) s = −∞ δ ( ρ − ρ (cid:48) ) K ( ρ, s ) × exp { is ( φ − φ (cid:48) ) } sin φ cos φ (cid:48) , (39) © , 000–000 Ya. N. Istomin, A. A. Philippov, and V. S. Beskin ε xx ( r , r (cid:48) ) = 1 − π ρ (cid:48) ∞ (cid:88) s = −∞ δ ( ρ − ρ (cid:48) ) K ( ρ, s ) × exp { is ( φ − φ (cid:48) ) } sin φ sin φ (cid:48) , (40)that provides the material relationship D i ( r ) = (cid:90) ε ij ( r , r (cid:48) ) E j ( r (cid:48) )d r (cid:48) . (41)It should be noted that the operator presented abovesatisfies the needed symmetry condition ε ij ( r , r (cid:48) , ω ) = ε ji ( r (cid:48) , r , − ω ) (42)(it is provided by the condition K ( r, s, ω ) = K ( r, − s, − ω )).As it is well-known (Kadomtsev 1965; Bornatici & Kravtsov2000), it is this symmetrical form of permittivity tensor thatis to be used for the calculation of components of the per-mittivity tensor ε ij ( ω, k , r ) ε ij ( ω, k , η → r ) = (cid:90) ε ij ( ω, ξ , η ) exp {− i k ξ } d r . (43)Here η = ( r + r (cid:48) ) / ξ = r − r (cid:48) . It is important that theabove tensor only describes correctly wave-particle interac-tion in inhomogeneous media with slowly varying parame-ters (Bernstein & Friedland 1984).Substituting now the kernel components, one can find ε xx ( ω, k , η ) = 1 − π (cid:90) d ξ exp {− i k ξ } | η − ξ / | × ∞ (cid:88) s = −∞ δ ( | η + ξ / | − | η − ξ / | ) K ( | η + ξ / | , s ) × exp { is ( φ − φ (cid:48) ) } sin φ sin φ (cid:48) , (44) ε xy ( ω, k , η ) = 12 π (cid:90) d ξ exp {− i k ξ } | η − ξ / | × ∞ (cid:88) s = −∞ δ ( | η + ξ / | − | η − ξ / | ) K ( | η + ξ / | , s ) × exp { is ( φ − φ (cid:48) ) } sin φ cos φ (cid:48) , (45) ε yx ( ω, k , η ) = 12 π (cid:90) d ξ exp {− i k ξ } | η − ξ / | × ∞ (cid:88) s = −∞ δ ( | η + ξ / | − | η − ξ / | ) K ( | η + ξ / | , s ) × exp { is ( φ − φ (cid:48) ) } cos φ sin φ (cid:48) , (46) ε yy ( ω, k , η ) = 1 − π (cid:90) d ξ exp {− i k ξ } | η − ξ / | × ∞ (cid:88) s = −∞ δ ( | η + ξ / | − | η − ξ / | ) K ( | η + ξ / | , s ) × exp { is ( φ − φ (cid:48) ) } cos φ cos φ (cid:48) . (47)In this equations, the angles φ and φ (cid:48) are the functions ofpolar angles of vectors η and ξ , α η , and α ξ sin φ = | η | sin α η + ( | ξ | /
2) sin α ξ | η + ξ / | , (48)cos φ (cid:48) = | η | cos α η − ( | ξ | /
2) cos α ξ | η − ξ / | . (49)As a result, integrals above are reduced to integration over ξ , which is perpendicular to η . On the other hand, the ex- pression for the delta-functions in (44)–(47) is the following: δ ( ... ) = δ ( θ − π/ | η + ξ / | − | η − ξ / | ) (cid:48) θ + δ ( θ + π/ | η + ξ / | − | η − ξ / | ) (cid:48) θ , (50)where θ is the angle between vectors η and ξ . So, the integra-tion over angles can be done easily. Finally, from the tran-sition η → r , one can obtain cos α η → cos α r = 1. Hence,according to (50) ( k ξ ) = k (cid:107) | ξ | , where k (cid:107) is the componentof the wave vector parallel to external magnetic field.The property of the absence of k ⊥ is very important, itprovides the same symmetry as it was in the case of homo-geneous medium: ε ij ( − ω, − k , − B , r ) = ε ji ( ω, k , B , r ) (Is-tomin 1994). This result differs from one obtained by Lyu-tikov at al. (1999). In this work the importance of transfor-mation (43) is neglected.Using finally the Taylor expansion over | ξ | and the re-duction of resonant denominator to delta-function, one canobtain: (cid:88) ( ... ) 1 ω | η + ξ / | /v φ − s → iπ (cid:90) ( ... ) δ (cid:20) s − ω ( | η | + | ξ | / / v φ (cid:21) d s. (51)As a result, one can write down ε xx = − i π e ω (cid:90) F (cid:48)(cid:48) ( κ ) v φ ω ∂f (0) ∂p φ d p φ , (52) ε xy = − ε yx = 8 π e ω (cid:90) F (cid:48) ( κ ) ρ / v / φ ω / ∂f (0) ∂p φ d p φ , (53) ε yy = − i π e ω (cid:90) F ( κ ) ρ / v / φ ω / ∂f (0) ∂p φ d p φ . (54)Here F ( κ ) = 1 π + ∞ (cid:90) exp { iκt + it / } d t, (55) κ = 2( ω − k (cid:107) v φ ) ω / v / φ ρ / , (56)prime means the derivative, and ρ is the curvature radiusof magnetic field.Due to high enough curvature radius of field lines in thepulsar magnetosphere, one can use the asymptotic behaviourof F ( κ ) for κ (cid:29) F ( κ ) ≈ iπκ + 2 iπκ + ... (57)After integration by parts, the final result is ε ij = − (cid:28) ω pl v (cid:107) γ ρ ˜ ω (cid:29) − i (cid:28) ω pl v (cid:107) γ ρ ˜ ω (cid:29) i (cid:28) ω pl v (cid:107) γ ρ ˜ ω (cid:29) − (cid:28) ω pl γ ˜ ω (cid:29) (58)Here by definition ˜ ω = ω − kv , and the brackets <> denoteboth the averaging over the particle distribution function f e + ,e − ( p φ ) and the summation over the types of particles: < ( ... ) > = (cid:88) e + e − (cid:90) ( ... ) f (0) e + ,e − ( p φ )d p φ . (59) © , 000–000 n the collective curvature radiation We see that the tensor above is just the BGI tensor, thatleads to instability of the so-called curvature plasma modes.In the limit ρ = ∞ this tensor, as expected, tends to the di-electric permittivity of a homogeneous plasma. The nonzerocomponents ε xy , ε yx and δε xx = ε xx − ε ij (58)for the finite curvature are due to nonlocal properties of theplasma response on the electromagnetic wave in curved mag-netic field. The parameter of nonlocality ( v (cid:107) / ˜ ω ) /ρ is theratio of the formation length of radiation to the curvatureradius. For vacuum ˜ ω (cid:39) ω/γ , and the length v (cid:107) / ˜ ω coincideswith the length of formation of the curvature radiation l f .It is important that the components ε xy = − ε yx and δε xx essentially change the wave polarization. The relationbetween E φ and E ρ of the wave electric field, following fromthe tensor of the dielectric permittivity (58), is( ε xy + n ρ n φ ) E φ + (cid:0) δε xx + 1 − n φ (cid:1) E ρ = 0 , (60)where n ρ and n φ are components of the dimensionless wavevector: n = k c/ω . For the tangent wave propagation (i.e.,for n ρ = 0) we have E φ (cid:39) ( δε xx /ε xy ) E ρ (cid:39) ( c/ρ c ˜ ω ) E ρ . Asa result, a wave can produce the negative work under theelectric particle current j φ , i.e., it can be excited. It is notso if δε xx = ε xy = 0 when E φ = 0. Thus, as it was shown above, the wave polarization[ E ρ ( ρ ); E φ ( ρ )] exp { isφ } containing one cylindrical harmonic s suggests only the Cherenkov mechanism of radiation. Inthe curvature radiation mechanism of one particle in vac-uum the generated wave consists of three harmonics s, s ± s -harmonic, means only the existence of the Cherenkov mecha-nism of the wave generation. In this case the centrifugal par-ticle drift places the significant role. Practically all curvatureeffects come only to this drift. And the Cherenkov resonanceon the drift motion produces small wave amplification inbetter case (Kaganovich & Lyubarsky 2010). Stronger mag-netic field produces less drift velocity and less Cherenkoveffect though the curvature of a particle motion does notdepend on the magnetic field strength at all. We thank A.V. Gurevich for his interest and support. Thiswork was partially supported by Russian Foundation for Ba-sic Research (Grant no. 11-02-01021).
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