Properties of magneto-dipole X-ray lines in different radiation models
aa r X i v : . [ a s t r o - ph . H E ] J a n MNRAS , 1–8 (2015) Preprint 1 August 2018 Compiled using MNRAS L A TEX style file v3.0
Properties of magneto-dipole X-ray lines in differentradiation models
G. S. Bisnovatyi-Kogan, , ⋆ Ya. S. Lyakhova, , Space Research Institute of Russian Academy of of Sciences, Profsoyuznaya 84/32, Moscow, 117997, Russia National Research Nuclear University MEPhI (Moscow Engineering Physics Institute),Kashirskoe Shosse 31, Moscow, 115409, Russia
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We compare polarization properties of the cyclotron, and relativistic dipole radiationof electrons moving in the magnetic field on a helix with ultra-relativistic longitudinaland non-relativistic transverse velocity components. The applicability of these modelsin the case of accretion onto a neutron star is discussed. The test, based on polarizationobservations is suggested, to distinguish between the cyclotron, and relativistic dipoleorigin of features, observed in X-ray spectra of some X-ray sources, among which theHer X-1 is the most famous.
Key words: keyword1 – keyword2 – keyword3
X-ray pulsar Hercules X-1 discovered in 1971 by the Uhurusatellite is one of the best studied X-ray source. Her X-1is the first source in which X-ray spectrum the line fea-ture in the 39-58 KeV energy range was observed, whichcould not be identified with any chemical element, and wassuggested to be a cyclotron line Tr¨umper (1978). This fea-ture was observed later in Tuelle, et al. (1984); Voges, et al.(1982); Ubertini, et al. (1980); Gruber, et al. (1980). Whenthis feature is interpreted as a cyclotron line, the magneticfield strength may be calculated from the non-relativisticformula B = m e cωe , (1)where ω is the cycle frequency of the electrons, identifiedwith the frequency of the observed X-ray feature, m e is themass of the electron, c is the light speed. In this case the mag-netic field strength should be of the order of (3 − × Gs. But as large as this value comes into conflict with sometheoretical reasonings among which the most important areconsideration of the interrelation between radio and X-raypulsars Bisnovatyi-Kogan and Komberg (1974), and simula-tion of the pulse variability during the 35-days cycle in obser-vations from the satellites ASTRON Sheffer, et al. (1992),Ginga and RXTE Scott, et al. (2000); Deeter, et al. (1998).Obscuration of X-ray beams during the 35 day cycle is oftenused to explain the periodic X-ray high-low state transitions ⋆ E-mail: [email protected] (GSBK) of Her X-1 during the accretion disk precession. If the ob-scuring material is the inner edge of the accretion disk, thenthe inner disk must be tilted out of the binary plane andbe precessing to produce periodically varying obscuration.In such a situation, occultation of the neutron star wouldoccur twice in each precession cycle, leading to the declinein flux, and termination of the main and short high states.This scheme was also extended Sheffer, et al. (1992) to ex-plain pulse profile evolution with a reflection of the light onthe off state by the inner edge of the accretion disk. Thevalue of the dipole magnetic field of the neutron star, de-termining the radius of the inner edge, coinciding with theradius of the Alfven surface, was estimated in this model as10 − Gs.Let us stress, that in this model the region where thenon-collision shock wave is formed is situated at the upperside of the accretion column, so it is separated in space fromthe region where the main X-ray flux is formed. Therefore,in connection with this model, there is no need to make anyprincipal changing in the standard model. Only the structureof the accretion column could be modified adjusting to thelower value of the magnetic field.The most reliable estimation of the magnetic field B S inneutron stars (dipole component) is obtained for radiopul-sars by measurements of a growth of their rotation periodat magneto-dipole losses. For single radiopulsars, forminga large group of about 2000 objects, this field is variesLorimer (2005) around 10 G. In addition to the mainbody of the objects on the diagram ( P, ˙ P ) there is a smallergroup of radiopulsars Lorimer (2005) with more rapid ro-tation and lower magnetic fields 10 − · . About200 of these pulsars are called ”recycled pulsars”, which c (cid:13) G. S. Bisnovatyi-Kogan, Ya. S. Lyakhova passed the stage of accretion in close binaries, when theygain a rapid rotational speed, and decrease their magneticfield Bisnovatyi-Kogan and Komberg (1974). Majority of re-cycled pulsars had low mass companions, remaining as awhite dwarf in the binary system, and lower magnetic fields10 − Gs, and few tens of pulsars are in the binary withanother neutron star, and magnetic field up to ∼ · ∼ M ⊙ , which ends its evolution as a white dwarf.Therefore, there is not surprising for the neutron star in thissystem to have a magnetic field in the range 10 − Gs.To solve the problem of discrepancy betweenthis estimation and the value following from thecyclotron interpretation (1), it was suggested inBaushev and Bisnovatyi-Kogan (1999), that the ob-served feature could be explained by the relativistic dipoleradiation of electrons having strongly anisotropic distri-bution function, with ultra-relativistic motion along themagnetic field lines, and non-relativistic motion across it.Such distribution function is formed when the accretion flowinto the magnetic pole of the neutron star is stopped in anon-collisional shock wave Bisnovatyi-Kogan and Fridman(1969), and a rapid loss of transversal energy in the strongmagnetic field leads to strongly anisotropic momentumdistribution Bisnovatyi-Kogan (1973).It is not possible for the moment to make a definitechoice between these two models. There are another modelsexplaining the change of X-ray beam during 35 day periodwithout obscuration of the beam by the inner edge of theaccretion disk (Postnov et al., 2013; Staubert et al., 2013).Therefore only observational criteria permit to make a choicebetween the models.In this paper we consider the problem of the observa-tional choice between the above mentioned models by mea-suring the polarization of the radiation in this X-ray feature.The relativistic dipole and cyclotron radiation have differentpolarization properties, so such measurements could solvethis long-standing problem. Such experiments could be per-formed on the Japanese satellite
Astro-H which launch isplanned for 2015, AstroH (2015). For description of differentways of X-ray polarization measurements see Pearce, et al.(2012); Kislat, et al. (2015), and references therein.
The cyclotron radiation is produced during a motion of non-relativistic electrons across a magnetic field direction. It isradiated in the form of the line with the energy ~ ω B , withthe cyclotron frequency ω B = eBm e c , ν = ω B π . (2)The electron is moving along the Larmor circle with theradius R L = mυ ⊥ , eB , (3)where the electron velocity υ ⊥ , is the component situated inthe plane perpendicular to the direction of the magnetic field in the frame connected with the Larmor circle. The electronis radiating also on the harmonic frequency ω nB = nω B .At υ ⊥ , ≪ c the strength of the harmonic lines is rapidlydecreasing with the number n . If also the total | υ | ≪ c ,the change of cyclotron frequency due to Doppler shiftingmay be neglected, and only the gravitational redshift in thegravitational field of the neutron star (not present in (1))should be taken into account for the magnetic field eval-uation. Taking into account only the radiation on the firstharmonic of the cyclotron frequency, we have it’s differentialangular emissivity W ( ϑ ) as Trubnikov (1961) W = e ω B υ ⊥ πc (cid:0) ϑ (cid:1) δ ( ω − ω B ) ergs · sterad · Hz ,υ ⊥ ≪ c, (4)and the total emissivity, after integration over the angle andfrequency, is: W tot = 2 e ω B υ ⊥ c ergs . (5)Expressions for the degrees of linear and circular polariza-tion, respectively, are written as Epstein (1973): ρ l = 1 − cos ϑ ϑ , (6) ρ c = 2 cos ϑ ϑ , ρ l + ρ c = 1 . (7)The cyclotron radiation of a single electron is totally po-larized, inducing the last equality. The cyclotron radiationalong the direction of the magnetic field is fully circularlypolarized and in the plane perpendicular to the magneticfield it’s fully linearly polarized. We shall use the subscript”0” for the frame, connected with the plane of the Larmorcircle where υ k = 0. Angular distribution of the emissivity:full W from (4), polarized linearly W l , and circularly W r of a cyclotron radiation are presented in Fig. 1. The linearand circular emissivities are determined as W l ( ϑ ) = W ρ l , W r ( ϑ ) = W ρ r , (8)where ρ l and ρ r are given in (6) and (7), respectively. Theangle ϑ = 0 corresponds to the direction of the magneticfield. Let’s consider an electron in the magnetic field, with thefollowing values of the velocity components in the laboratoryframe υ k ≃ c, γ k = 1 r − υ k c ≫ ,υ ⊥ ≪ c s − υ k c = cγ k . (9) MNRAS , 1–8 (2015) roperties of magneto-dipole lines Figure 1.
Angular distribution of CR polarization components(a) and angular dependence of the linear and circular polarizationdegrees (b). Arbitrary units are used.
The trajectory of the electron is helical, with the helix stepsignificantly larger than it’s radius (see Fig. 2).The radiation provided by such system is calledZheleznyakov (1997) Relativistic Dipole (RDR). The proper-ties of RDR have been considered in detail in Epstein (1973).The calculations of the angular distributions of RDR emis-sivity power, and both types of polarization in the laboratoryframe, where the electron is moving along the magnetic fieldto the observer with the velocity υ k , may be calculated bymaking Lorentz transformation in (4),(6),(7). The angle ϑ and velocity υ ⊥ in the Larmor circle frame are connectedwith the angle ϑ and velocity υ ⊥ in the laboratory frame as( β k = υ k /c )sin ϑ = sin ϑ q − β k − β k cos ϑ , cos ϑ = cos ϑ − β k − β k cos ϑ ,υ ⊥ = γ k υ ⊥ . (10) Figure 2.
The RDR trajectory: electron moves on helix alongthe magnetic field; h is the helix step. Expressions for the linear and circular polarization degreesin the laboratory frame are obtained from (6),(7), with ac-count of (10), as ρ l = 1 − (cid:16) cos ϑ − β k − β k cos ϑ (cid:17) (cid:16) cos ϑ − β k − β k cos ϑ (cid:17) , ρ c = 2 cos ϑ − β k − β k cos ϑ (cid:16) cos ϑ − β k − β k cos ϑ (cid:17) . (11)It follows from (4), (10) that RDR radiation is emitted in asmall angle ( ϑ . /γ k ), along the magnetic field direction.We have, by definition, β k = 1 − (1 /γ k ). For small ϑ , andlarge γ k we have the following expansions β k ≈ − γ k − γ k , cos ϑ ≈ − ϑ ϑ , cos ϑ ≈ − ϑ + ϑ , (12)It is convenient Bordovitsyn (1999), to introduce a variable ψ = γ k ϑ. (13)With account of (13) we obtain from (12) the expressions1 − β k cos ϑ ≈ γ k (cid:0) ψ (cid:1) , cos ϑ − β k ≈ γ k (cid:0) − ψ (cid:1) . (14)The emission of RDR is monochromatic in the laboratoryframe in any given direction, with the frequency and po-larization depending on the angle θ . The angular frequencydistribution is obtained from the relations for Doppler ef-fect Landau, Lifshitz (1975) (see also Epstein (1973)), which, MNRAS , 1–8 (2015)
G. S. Bisnovatyi-Kogan, Ya. S. Lyakhova with account of (14), are written for the time interval dt andthe frequency ω , as dt = dt q − β k , (15) ω = ω B q − β k − β k cos ϑ , γ k ω B ≥ ω ≥ ω B γ k at 0 ≤ ϑ ≤ π. Approximately we have ω ≈ γ k ω B ψ , ψ = 2 γ k ω B ω − . (16)Angular dependencies of the polarization degrees in the lab-oratory frame from (11), with account of (13),(15) are writ-ten as Epstein (1973): ρ l ( ω ) ≈ ψ ψ = ω (cid:0) γ k ω B − ω (cid:1) ω − γ k ωω B + 2 γ k ω B , (17) ρ c ( ω ) ≈ − ψ ψ = 2 γ k ω B (cid:0) ω − γ k ω B (cid:1) ω − γ k ωω B + 2 γ k ω B . (18)The relative graphics of angular dependencies are presentedon Figs 3 – 6.The differential power of the radiation in the unity ofthe solid angle Ω, with d Ω = sin ϑdϑdϕ , time t , and fre-quency is obtained from (4), with account of (10), (15), andrelations d Ω = d Ω d cos ϑ d cos ϑ = d Ω 1 − β k (1 − β k cos ϑ ) , (19)We have than from (4), using (10),(15), the expression forthe differential emissivity in the laboratory frame as (seeEpstein (1973)) W = W (1 − β k ) / (1 − β k cos ϑ ) = e ω B υ ⊥ πc (cid:20) ϑ − β k ) (1 − β k cos ϑ ) (cid:21) × (1 − β k ) / (1 − β k cos ϑ ) δ [ γ k ω (1 − β k cos ϑ ) − ω B ] . (20)After transformation of δ -function we have W = e ω B υ ⊥ πc γ k (cid:20) ϑ − β k ) (1 − β k cos ϑ ) (cid:21) (21) × (1 − β k ) / (1 − β k cos ϑ ) δ (cid:20) ω − ω B γ k (1 − β k cos ϑ ) (cid:21) . Approximately we have W ≈ e ω B γ k υ ⊥ πc ψ (1 + ψ ) δ (cid:18) ω − γ k ω B ψ (cid:19) . (22)Using (16) we obtain in the laboratory frame Epstein (1973)) W ≈ e υ ⊥ ω πc γ k ω B − ωγ k ω B + ω γ k ω B ! (23) Figure 3.
Angular distribution of RDR polarization componentsfor different values of Lorentz parameters: 3.1) γ k = 1 .
1; 3.2) γ k = 1 .
5. Arbitrary units are used. × δ (cid:18) ω − γ k ω B ψ (cid:19) ergs · sterad · Hz . The spectral distribution of RDR is obtained after integra-tion over d Ω W ( ω ) = Z Ω W d
Ω ergs · Hz , (24)and the total emissivity W tot = Z ω max W ( ω ) dω ergs . (25)Taking into account (13),(16), we have the relations for δ - MNRAS , 1–8 (2015) roperties of magneto-dipole lines Figure 4.
Angular dependence of the linear and circular polar-ization degrees for different values of Lorentz parameters: 4.1) γ k = 1 .
1; 4.2) γ k = 1 .
5. Arbitrary units are used. function as2 πδ (cid:18) ω − γ k ω B ψ (cid:19) sin θdθdω ≈ πδ (cid:18) ω − γ k ω B ψ (cid:19) dθ dω (26)= π ψ γ k ω δ (cid:18) ψ − γ k ω B ω (cid:19) d ( ψ + 1) dω, After integration in (24), with account of (26), we obtain W ( ω ) = e υ ⊥ ω c γ k ω B − ωγ k ω B + ω γ k ω B ! × γ k ω B ω (27)= e υ ⊥ ω c γ k − ωγ k ω B + ω γ k ω B ! . Figure 5.
Angular distribution of RDR polarization componentsfor different values of Lorentz parameters: 5.1) γ k = 3 .
0; 5.2) γ k = 10 .
0. Arbitrary units are used.
The total emissivity, defining the rate of the energy loss of aparticle, is obtained after integration of (27), with ω max =2 γ k ω B from (15) as W tot = Z ω max W ( ω ) dω = 2 e υ ⊥ ω B c ergs , (28)in accordance with (5). MNRAS , 1–8 (2015)
G. S. Bisnovatyi-Kogan, Ya. S. Lyakhova
Figure 6.
Angular dependence of the linear and circular polar-ization degrees for different values of Lorentz parameters: 6.1) γ k = 3 .
0; 6.2) γ k = 10 .
0. Arbitrary units are used.
We consider here, that the angular size of the aperture of theX-ray detector ∆ ϑ X is much larger than the characteristicbeam width of RDR ϑ RDR = 1 /γ k , so that ∆ ϑ X ≫ /γ k , γ k ≫
1. If the angular size of the hot spot on the neutronstar magnetic poles ϑ hs ≫ ϑ RDR , than the registered X-rayRDR radiation is coming from the part of the hot spot, andis lasted during the time τ RDR ≈ P ϑ hs / π . It is decreasingabruptly outside this time interval. Note that in the caseof ordinary cyclotron radiation its intensity is decreasingsmoothly because of almost isotropic radiation diagram ∼ (1 + cos ϑ ), according to (4). Let us mention 3 mechanisms of the line widening in themagneto dipole radiation.1) A variable magnetic field B in the region of theline formation. This mechanism is working in the cyclotronradiation of nonrelativistic electrons, emitting the line atfrequency ω = ω B = eBm e c , as well as in RDR where thefrequency of radiation in the frame of the Larmor circle ischanging similarly.Two other mechanisms are characteristic only to theRDR.2) The distribution of electrons over theparallel momentum p k may be presented asBaushev and Bisnovatyi-Kogan (1999) f e ( p k ) = n e √ πσ exp (cid:18) − ( p k − a ) σ (cid:19) . (29)Such distribution of electrons leads to widening of the lineas Baushev and Bisnovatyi-Kogan (1999)∆ ω = ω B σm e c . (30)For narrow momentum distribution of electrons σ ≪ a wehave∆ ω ≪ ω B am e c ≈ ω B γ k . (31)In this case the line width may be very narrow. If σ < m e c ,the width of the line will be∆ ωω < γ k . (32)3) In the case of large γ k ≫ x = ωγ k ω B , the frequency distribu-tion in the registered signal (27) may be written as W ( ω ) ∼ ˜ W ( x ) = x (1 − x + x , γ k < x < . (33)Here ˜ W max = 2 at x max = 2, the median emissivity W = 1is reached at x med ≈ .
52, the 50% and 90% of emissionis concentrated inside 1 . < x <
2, and 0 . < x < WW tot = R x , x (1 − x + x ) dx R x (1 − x + x ) dx = 0 . , . x ≈ .
5. The effective width of theline is about δω ≈ . ω max . The form of the line on thephoton counts figure n γ ( ω ) follows the law n γ ( x ) = 1 − x + x , at 12 γ k < x < , (35)with a smooth slope at lower frequencies and abrupt brake MNRAS , 1–8 (2015) roperties of magneto-dipole lines Figure 7.
The frequency distribution of energy in the RDR line.
Figure 8.
The frequency distribution of the photons. The sym-metrical form of the profile corresponds to the symmetry of theangular distribution of the radiation (see Fig. 1). at x = 2, ω = 2 γ k ω B . Stress, that this form of the line ispresent for monoenergetic electron beam with γ k equal forall electrons. This type of widening takes place only in theRDR mechanism, and the form of the magneto-dipole linemay be used for distinguishing between the cyclotron andRDR mechanisms. Note that in presence of other wideningmechanisms the line may become even wider, but should pre-serve characteristic feature of this mechanism. The form ofthe line ˜ W ( x ) and the frequency distribution of the photonsare shown on Figs 7 and 8. Linear W l ( ω ) and circular W c ( ω ) parts of the total radiation,using (17), (18), (27), may be expressed in the following way W l ( ω ) = W ( ω ) ρ l ( ω ) = e v ⊥ ω B c γ k x (1 − x ) − x + x , (36) W c ( ω ) = W ( ω ) ρ l ( ω ) = e v ⊥ ω B c γ k x (1 − x ) − x + x . (37)Here x = ωγ k ω B , 0 < x <
2. It is easy to show that W l ( ω ) + W c ( ω ) = W ( ω ) from (27). The linear polarization of thetotal radiation in the beam W l is obtained by integrationover the frequency, it gives W l = Z W l ( ω ) dx = e v ⊥ ω B c γ k Z x (1 − x ) − x + x dx = (38)= ( π −
83 ) e v ⊥ ω B c γ k . The linearly polarized part of the radiation in the beam
RDR l is defined as RDR l = W l W tot = 34 Z x (1 − x ) − x + x dx = 34 ( π −
83 ) ≈ (39) ≈ · . ≈ . . The circularly polarized part of the beam radiation has dif-ferent sign in the wavelength intervals ω B γ k < ω < γ k ω B (0 < x < ω B γ k < ω < γ k ω B (1 < x < RDR c (0 < x <
1) is determined as
RDR cl = W c W tot = "Z x (1 − x ) − x + x dx / (cid:20)Z x (1 − x + x (cid:21) = (ln 2 − π /
724 = 0 . . ≈ . . (40)The circularly polarized part of the radiation at higher fre-quencies RDR c (1 < x <
2) has an opposite sign of polar-ization, and is determined as
RDR c = W c W tot = "Z x (1 − x ) − x + x dx / (cid:20)Z x (1 − x + x (cid:21) = (3 − ln 2 − π / . . ≈ . . (41) MNRAS , 1–8 (2015)
G. S. Bisnovatyi-Kogan, Ya. S. Lyakhova
The analysis of two model of the magneto dipole mecha-nism of the line formation had shown important features,by measuring of which the cyclotron model may be dis-tinguished from the RDR model with strongly anisotropicelectron distribution. The angular distribution in these mod-els is very different, quasi-isotropic in the cyclotron model,and strongly anisotropic in RDR. Nevertheless, in observingthese, usually rather weak lines, it is hardly possible to ob-tain a definite answer about the angular distribution, dueto many contaminating factors. Different form of the line,and different observed light curves, predicted by theory forthese lines could help more, when measured, but for weaklines it seems also very difficult. The most distinct differencebetween the cyclotron and RDR mechanisms of the line for-mation may be seen in their polarization features. The mea-surements of the hard X-ray polarization are discussed formore than 40 years, but still there is no space mission forthese measurements. The linear X-ray polarization which ismeasured should be close to zero for the cyclotron radiationfrom the hot magnetic pole, while the radiation produced inRDR model should have about 35% of the linear polarizationin the line, what is sufficiently well distinguished difference,which could be taken into account in the construction of thehard X-ray polarimeter. The first source with the detectedmagnetic dipole line Her X-1 is still the best target for thisinvestigation.The line emitted by non-relativistic electrons in themagnetic field has the same cyclotron frequency. Its harmon-ics are highly suppressed at kT ≪ m e c , what is expectedin the accretion disk and in the accretion column. The ob-served line width originating from photons of the relativisticdipole radiation coming from different angles is forming onebroad line, contrary to separate harmonics in the cyclotronmodel. The second harmonic in the case of relativistic dipoleradiation should form another line of almost the same widthat double energy. There is still not clear wether the second”cyclotron” harmonics is present in the X-ray spectrum ofHer X1 (Tr¨umper at al., 1978; Enoto et al., 2008; F¨urst etal., 2013).The influence of the line broadening due to the distribu-tion over the parallel electron momentum is considered byBaushev and Bisnovatyi-Kogan (1999). According to (32),the broadening is small when the scattering momentum isnon-relativistic. With increasing of σ over m e c , the broaden-ing increases, and the resulting line broadening is the com-bination of the angular and scattering action. Nevertheless,it is very important that for any scattering in a parallel di-rection the parameters of the polarization do not change forany large γ k , which are considered in this model, and thecriteria for the choice between models remains valid. ACKNOWLEDGEMENTS
This work was partially supported by the Russian Foun-dation for Basic Research Grant No. 14-02-00728 and theRussian Federation President Grant for Support of LeadingScientific Schools, Grant No. NSh-261.2014.2. The work ofGSBK was partially supported by the Russian Foundationfor Basic Research Grant No. OFI-M 14-29-06045. The workof YaSL was partially supported by the Dynasty Foundation.
REFERENCES
AstroH (2015) Astronomical project
Astro-H : http://astro-h.isas.jaxa.jpBaushev A.N., Bisnovatyi-Kogan G.S. (1999) Astronomical Re-ports, 43, 241Bisnovatyi-Kogan G.S. (1973) AZh, 50, 902Bisnovatyi-Kogan G.S., Fridman A.M. (1969) AZh, 46, 721Bisnovatyi-Kogan G.S., Komberg B.V. (1974) AZh, 51, 373Bordovitsyn V.A. ed. (1999)
Synchrotron radiation theory and itsdevelopment , Singapour, World ScientificDeeter J.E., Scott D.M., Boynton P.E., Miyamoto S., KitamotoS., Takahama S., Nagase F. (1998) ApJ, 502, 802Enoto T., Makishima K., Terada Y., Mihara T., Nakazawa K.,Ueda T., Dotani T., Kokubun M., Nagase F., Naik S., SuzukiM., Nakajima M., and Takahashi M. (2008) Publ. Astron. Soc.Japan 60, S57Epstein R. (1973) ApJ, 183, 593F¨urst F., Grefenstette W.B., Staubert R., Tomsick J.A., BachettiM., Barret D., Bellm E.C., Boggs S.E., Chenevez J., Chris-tensen F.E., Craig W.W., Hailey C.J., Harrison F., KlochkovD., Madsen K.K., Pottschmidt K., Stern D., Walton D.J.,Wilms J., and Zhang W. (2013) ApJ, 779, 69Gruber D.E., Matteson J.L., Nolan P.L., Knight F.K., BaityW.A., Rothschild R.E., Peterson L.E., Hoffman J.A., Scheep-maker A., Wheaton W.A., Primini F.A., Levine A.M., LewinW.H.G. (1980) ApJ.(Letters), 240, L27Kislat F., Beilicke M., Guo Q., Zajczyk A., Krawczynski H. (2015)Astropart. Phys. 64, 40Landau L.D., Lifshitz E.M. (1975)
The classical theory of fields
Radiation in the astrophysical plasma ,Moscow. Yanus-K.(in Russian)This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000