Linear Transformation and the Escape of Waves from Pulsar Magnetospheres

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1 Astronomy Letters, Vol, No 4, 996, pp Translated from Pis ma v Astronomihesii Zhurnal, Vol, No 7 8, 996, pp Original Russian Text Copyright 996 by Blioh, Lyubarsii Linear Transformation and the Esape of Waves from Pulsar Magnetospheres K Yu Blioh and Yu E Lyubarsii Radio Astronomy Institute, Urainian Aademy of Sienes, Kharov, Uraine Kharov State University, Kharov, Uraine Reeived July 9, 995; in final form, November 3, 995 Abstrat Two-stream instability in pulsar magnetospheres generates longitudinal waves, whih fall in the superluminal part of the dispersion urve due to stimulated sattering A superluminal longitudinal wave propagating in a urved magneti field turns into an eletromagneti wave and may esape from plasma in the form of an ordinary transverse wave However, this wave may go to the subluminal part of the dispersion urve, where it ultimately deays through Landau damping, due to linear transformation Linear transformation of waves in an ultrarelativisti eletron positron plasma embedded in a superstrong magneti field is onsidered The transformation is shown to our only in the ase of propagation at very small angles to the magneti field; this proess annot hinder the esape of waves from pulsar magnetospheres INTRODUCTION An investigation into the nature of pulsar radio emission has required the study of eletromagneti waves propagating in an ultrarelativisti eletron positron plasma embedded in a strong magneti field An extensive analysis of the properties of these waves and referenes to previous papers an be found in Voloitin et al (985), Arons and Barnard (986), Lominade et al (986), and Lyubarsii (995) Figure shows the struture of the dispersion urves In the ase of propagation along the magneti field, there are purely longitudinal and purely transverse waves; their dispersion urves interset at the point where Here, 3ne γ , m where γ is the Lorent fator of the plasma; n is the eletron density, whih is taen to be equal to the positron density; and m and e are the mass and harge of the eletron, respetively In the ase of propagation at some angle to the magneti field, the dispersion urves diverge in suh a way that one of them lies ompletely in the domain of superluminal phase veloities >, while the other lies in the subluminal domain < If the angle between the magneti field and the diretion of wave propagation is small, then the dispersion urve of the superluminal wave is lose to that of the longitudinal wave propagating along the field at < and to the dispersion urve of the transverse wave at > Therefore, the polariation of the superluminal wave will be nearly longitudinal at < and nearly transverse at > Conversely, the subluminal wave (it is alled an Alfvén wave, beause at low frequenies it transforms to a magnetohydrodynami Alfvén wave) has a nearly transverse polariation at < and a nearly longitudinal polariation at > The fat that the dispersion urves of waves propagating obliquely to the magneti field ouple the longitudinal wave with the transverse wave is of great importane for the esape of waves from plasma If the wave is propagating in the diretion of dereasing plasma density and the harateristi sale of density variations is large ompared to the wavelength, then the wave / / Fig Dispersion urves of waves propagating in an ultrarelativisti plasma in an infinitely strong magneti field The dashed lines are the urves for the waves propagating along the magneti field; the solid lines are the urves for the waves propagating at an angle to the magneti field See text /96/4-48$ 996 åäàä ç ÛÍ /Interperiodia Publishing

2 LINEAR TRANSFORMATION AND THE ESCAPE OF WAVES 483 number varies in suh a way that the dispersion equation at eah point is satisfied In Fig, the point representing the wave is shifted upward along the dispersion urve The phase veloity ---- of the longitudinal wave propagating exatly along the magneti field dereases until it beomes equal to the plasma veloity, and the wave is then absorbed through Landau damping If, however, the wave is propagating even at a small angle to the magneti field, its dispersion urve does not interset the light one Therefore, the superluminal longitudinal wave will not experiene the Landau damping; rather, rather it goes to that part of the dispersion urve where its polariation tends to the transverse polariation, and where the dispersion law tends to the vauum law As a result, this wave an freely esape from the plasma in the form of an ordinary transverse eletromagneti wave Conversely, an Alfvén wave propagating in the diretion of dereasing plasma density will ultimately beome longitudinal, its phase veloity will derease, and the Landau damping will eventually mae further propagation of the wave impossible Thus, within the framewor of geometrial optis, the propagation of waves even at an arbitrarily small angle to the magneti field differs qualitatively from wave propagation exatly along the field However, at suffiiently small angles, the dispersion urves approah eah other losely near the point, and the approximation of geometrial optis may be violated: linear transformation of waves beomes possible [see, eg, Ginburg (967), Zhelenyaov (977), and Zhelenyaov et al (983)] It is this proess that maes the transition from finite to ero angle ontinuous At suffiiently small angles, the longitudinal superluminal wave goes over near the point to the subliminal branh, ie, the propagation ours in the same way as that for ero angle Note that, in the absene of an external magneti field, the longitudinal wave propagating in an isotropi medium without transformation remains longitudinal and subsequently deays; only as a result of transformation an it beome transverse and esape from the plasma In the ase of an anisotropi medium with an infinitely strong external magneti field, the wave without transformation traveling along dispersion urve beomes nearly transverse and freely esapes from the region in question In the ase of strong interation, the wave goes over to dispersion urve, remaining nearly longitudinal and subsequently deaying Thus, whereas in an isotropi medium the wave polariation hanges from longitudinal to transverse, in the anisotropi medium under onsideration the transformation, ie, the transfer from one branh of the dispersion urve to the other, in ontrast, results in the longitudinal (more preisely, nearly longitudinal) wave remaining longitudinal; if there is no transformation, the wave, remaining on the same dispersion urve, hanges its polariation In this paper, we will onsider wave transformation in an effort to find onditions for the esape of waves from the magnetosphere In pulsar magnetospheres, the longitudinal waves an be generated by two-stream instability [see, eg, Cheng and Ruderman (977), Egorenov et al (983), Asseo et al (98, 983), Usov (987), Ursov and Usov (988), Lyubarsii (993), and Weatherall (994)] True, it is mainly waves with a subluminal phase veloity that are generated in this proess; however, stimulated sattering transfers them to the domain of superluminal phase veloities (Lominade et al 979; Lyubarsii 996), with their frequeny tending to /, whih is the natural plasma frequeny shifted by a fator of γ due to the Doppler effet We will show that superluminal longitudinal waves virtually always esape from pulsar magnetospheres in the form of ordinary transverse waves BASIC EQUATIONS IN THE CASE OF A ONE-DIMENSIONAL INHOMOGENEITY In the region of a possible interation of waves, the pulsar magnetosphere is an eletron positron plasma in a dipole magneti field that is so strong that motion of partiles aross the field lines may be negleted The plasma moves with a onstant ultrarelativisti veloity v orresponding to the Lorent fator γ In pulsars, waves are generated in a narrow region near the magneti axis and emitted virtually along the magneti field Sine the plasma moves along the diverging field lines, its density falls off along the ray of the wave emerging from the magnetosphere, maing the effet of linear transformation near possible Assuming that the dimensions of this region are small ompared to the harateristi sale of the magnetosphere and noting that the field lines near the magneti field are nearly parallel, we first onsider a simpler problem Let the magneti field be uniform and direted along the axis of the Cartesian oordinate system We also assume that the densities of eletrons and positrons depend in the steady state only on : n n + n() (the effets assoiated with a nonuniform magneti field and the dependene of the density will be onsidered in setion 5) The eletromagneti field and the motion of partiles in a plasma are desribed by Maxwell s equations, the equations of motion, and the ontinuity equation: urlb -- E j; urle t B, () t dp ± ± ee, () dt ρ j , (3) t where j is the urrent density; ρ is the harge density; and p + and p are the momenta of the positrons and ASTRONOMY LETTERS Vol No 4 996

3 484 BLIOKH, LYUBARSKII eletrons, respetively Assuming that the wave amplitudes are suffiiently small, we an write the veloities and densities of the eletrons and positrons as v ± v + ṽ ± ( r, t); n ± n () + ñ ± ( r, t) and solve equations () and (3) in the linear approximation for ṽ ±, ñ ±, E, and B For example, given j ev( ñ + ñ ) + en( ṽ + ṽ ); ρ eñ ( + ñ ), and onsidering the differene of equations (), we obtain d ---- ( j ρv ) ( j ρv ) v dt ---dn n d ne E mγ 3 (4) Now let a monohromati wave with frequeny propagate in the plasma Sine the system is homogeneous in x, y, and t, the quantities E, B, ρ, j may be onsidered proportional to e i ( x x+ y y t), where the proportionality oeffiients are funtions of Without loss of generality, we may set y From the system of equations () and (4), we then obtain db -- y i, (5) d ---E x i x B y i ---E 4, (6) j de x i, (7) d x E i ---B y v dj ( ρv) j ρv d ( ) i v L dj ---- iρ, d (9) where dn is a quantity harateriing the L n d degree of nonuniformity, and the notation introdued in setion is used In our ase, is a funtion of In the subsequent analysis, we will assume the following ondition to be fulfilled: λ ---- () L --- n --dn -----, d whih implies that the waves in the medium under onsideration differ only slightly from normal waves in a homogeneous medium E, (8) 6γ 4 System (5) (9) redues to the system of two seondorder differential equations d B y, () d x i By x j d j d i i v ---- dj i + L j d v 4γ 4 v L v () + i x B 6γ 4 v y, whose four independent solutions orrespond to four possible types of waves in the given system However, we are interested in the interation of only two types of waves with the dispersion urves shown in Fig If the angle ϑ between the wave vetor and the diretion of the magneti field is small, then the refrative indies of these waves at frequenies near are lose to unity, ie, N,, ( ϑ, ) Therefore, in the ase of a suffiiently wea nonuniformity, the quantities j and B y an be written as j a ()e i --- ; By a ()e i e i ( x x t) (3) (the fators, whih from here on anel out, are omitted), assuming that a, a are slow funtions of, ie, µ da i a i d (4) d a Disarding the terms of the order of i, we then have d i ---da d i ---da (6) d x a i 4 x a Here, we have also disarded the small terms ontaining dn by assuming that the transformation was L n d determined by the terms that expliitly ontain n() We an show that in our problem this is admissible, substituting solutions for a, a (obtained below) into () and () Now system (5), (6) has only two independent solutions whih desribe the evolution of the two waves of interest to us We negleted the influene of the remaining two waves by dropping the seond derivatives This an be done if the refrative indies differ from unity for waves and (Fig ) muh less than for the other waves: N, N 3, a i a x, (5) 4γ v 6γ ASTRONOMY LETTERS Vol No 4 996

4 LINEAR TRANSFORMATION AND THE ESCAPE OF WAVES 485 By analying the dispersion equations for waves in the medium under onsideration at n() onst, we an show that for this inequality plaes a onstraint on the angle ϑ: ϑ γ --, (7) whih we will assume to be fulfilled We will show below that linear transformation in pulsars an be signifiant only for waves that satisfy ondition (7) Substituting a from (6) into (5), we obtain d a d i -- x δ da a γ d 8γ, (8) v where we used the ondition and γ introdued the notation δ() () Aording to the above assumptions, δ (9) 3 TRANSITION TO THE HOMOGENEOUS CASE Let us first onsider a homogeneous medium, δ() onst In this ase, we see solutions in the e i ---ξ form a If onditions (7) and (9) are fulfilled, we derive from (8) δ, ± 4γ ξ () Returning to the variables (3), whih depend on as e i in a homogeneous medium, we find the dispersion relations for the waves under onsideration: N δ ± 4γ () This formula desribes the dispersion urves, shown in Fig, near the transformation region ~, with the plus and minus signs orresponding to waves and, respetively Let us onsider the polariation of these waves x δ x 4γ 8 γ, ξ δ x 4γ 8 γ From equations (5) and (7), we readily obtain E ε x, E, x Substituting the dispersion relations () for δ γ x, ie, suffiiently far from the point, near γ whih the polariation varies greatly, we find that if δ < (ie, < ), then ε and ε If δ > (ie, > ), then ε and ε Thus, we arrive at the result that was already mentioned in setion : for < wave is nearly transverse and wave is nearly longitudinal, and, onversely, for > wave is nearly longitudinal and wave is nearly transverse When passing through a small region δ, waves and reverse their polariation 4 TRANSFORMATION OF WAVES Now let us return to equation (8), desribing the nonhomogeneous ase Using the fat that the sale of the region of wave interation is small ompared to L, we an replae δ() with a linear funtion δ , where is the value of at whih δ( ) L Introduing a new dimensionless variable s x ( ) x γ L () γ in plae of, we redue equation (8) to a'' ifa' + a, (3) where the prime denotes differentiation with respet to s, F Rs, R onst (4) L x The solutions of equation (3) an be expressed in terms of the Whittaer funtions of an imaginary argument; however, it is more simple to use Zwan s method (see, eg, Zaslavsii et al 98) to solve the transformation problem Note first that equation (3) has an exat first integral Q a' + a onst, (5) whih expresses the law of onservation of the total energy in the transformation of waves Further, for F, ie, far from the interation region, the solutions an be sought in the WKB approximation in the form a e iψ Using the method of perturbations, we, x ξ, ASTRONOMY LETTERS Vol No 4 996

5 486 BLIOKH, LYUBARSKII The law of onservation of energy (5) now taes the form an find two independent solutions in the first approximation for the small parameter -- : F (6) (7) The general solution of the equation for F an then be written in the form a Af + Bg, (8) where f Ψ F ds + ilnf+ O -- F F ; Ψ s O F -- d + F s -- F exp i F ds F s g exp i ds e i F s Fig The diretion of going around the origin in the omplex ŝ plane e irs Rs ln s R ( ln s ) R ; (9) (3) If the summands in (8) are of different orders, then we must retain only the greater (in absolute value) summand lest the auray of the approximations be exeeded Formulas (9) and (3) desribe waves in the rayoptis approximation, orresponding to replaement of the exponential fator e i in the ase of a homogeneous medium by the fator exp [ i () d] for a wealy inhomogeneous medium We an verify this by using the dispersion relations () Solution f orresponds to a nearly longitudinal wave, ie, wave for < and wave for > ; solution g orresponds to a nearly transverse wave, ie, wave for < and wave for > In what follows, wave transformation is said to tae plae if wave transforms to wave or wave transforms to wave ; in this ase, the polariation varies only slightly, and wave f remains wave f, while wave g remains wave g Q A + B (3) As was noted above, the transformation of waves ours in the region F However, the transformation oeffiient T is determined by the amplitudes A and B for F and F : T A ex A + B B en B ex A + B A en (3) where by the amplitudes at the entrane to the interation region and at the exit from it we mean the amplitudes for F and F, respetively To lin the solutions in the regions F and F without passing through the region F, we pass over to the omplex plane ŝ and go around the origin of oordinates at a suffiient distane (Fig ) The solution now taes the form a  fˆ + Bˆ ĝ, where  and Bˆ are onstant oeffiients, and fˆ and ĝ are funtions of the omplex variable ŝ that are given by the following relations: ( lnŝ) R, fˆ e irŝ Rŝ e i lnŝ R (33) ĝ (34) We hoose the ut along the positive diretion of the real axis Let there be initially only the longitudinal wave for F, ie, B en, Q A en, and fˆ en f en and  en A en on the real axis We will go around F ŝ in the lower half-plane (Fig ) In that ase, as an be seen from (33), fˆ first exponentially dereases, then inreases, and then again dereases Beause of the phase differene in the logarithm, the real-valued fator R e grows in fˆ, and, returning to the funtion f defined by (9) on the real axis, we obtain fˆ ex e R f ex Sine  is onstant, the hange of the real-valued fator of the funtion f, ie, of the amplitude A, will be A ex e A en R We annot obtain B ex by going around the origin in the lower half-plane, beause in the fourth quadrant g is lost against the baground of exponentially large f and, ASTRONOMY LETTERS Vol No 4 996

6 LINEAR TRANSFORMATION AND THE ESCAPE OF WAVES 487 goes beyond the limits of the auray of the problem However, invoing the law of onservation of energy (3), we obtain B ex Q A ex A R en e Similarly, if there is only the transverse wave for F (A en, Q B en ), then going around the origin of oordinates in the upper half-plane gives B ex R e, B en in omplete agreement with (3) Thus, we an finally assert that for R > the longitudinal wave transforms into the longitudinal one A ex A en [orresponding to a transition from the dispersion urve to the dispersion urve (Fig )] For R <, the longitudinal wave with exponential auray transforms into the transverse wave (ie, it remains on the dispersion urve ) Noting that x x ϑ, we obtain from (4) R λ L ϑ Hene we onlude that the transformation taes plae only if the waves propagate at an angle ϑ ---- λ L (35) to the magneti field Thus, if ondition (35) is fulfilled, the superluminal wave emitted with frequeny < and propagating in the diretion of density derease turns into a subluminal wave and is ultimately absorbed due to Landau damping; onversely, the subluminal wave turns into a superluminal wave and esapes from the plasma in the form of a transverse eletromagneti wave If the ondition that is the reverse of (35) is fulfilled, then there is no transformation, and in this ase the superluminal wave esapes from the plasma, while the subluminal wave is absorbed Let us also estimate the sie δ of the region where the transformation of waves ours We assume the waves to be independent for F, and, hene, the interation taes plae within the region F Using () and (4), we obtain δ ~ L ϑγ (36) The requirement δ L leads to ondition (7), whih we have already used 5 VARIATIONS IN THE ANGLE BETWEEN THE MAGNETIC FIELD AND THE WAVE VECTOR DURING WAVE PROPAGATION IN THE MAGNETOSPHERE When we onsidered wave transformation in setion 3, we assumed that the angle ϑ between and the diretion of the magneti field does not vary, ie, if the field is uniform, the rays are straight lines In fat, this is not the ase The angle ϑ varies due to ray bending resulting from nonuniformity of the magneti field in pulsar magnetospheres and from the existene of a transverse gradient in the plasma density, whih we will desribe by the quantity L x : ---- (37) L x n n x In addition, even if the waves propagated along straight lines, ϑ would vary due to the urvature of the magneti-field lines Let us begin with a qualitative analysis of the effets assoiated with the transverse gradient, assuming that the magneti field is uniform Later, we will onsider a more general ase Note first that the transverse gradient in the absene of the longitudinal gradient ---- annot produe the effet of wave transformation Indeed, in this ase the dispersion equation is written in the WKB approximation as x x (,, x), with and being onstant along the ray due to the homogeneity in t and Thus, it is now onvenient to desribe the wave in terms of the motion of a point in the -----, x plane along the dispersion urves Taing the form of the dispersion relations in the medium under onsideration () into aount, we an readily see that there is always only one branh of the dispersion urve in the -----, x plane Consequently, the transformation, ie, the transition of the point from one urve to the other, annot our For the transformation to our, a hange either in or in is needed and, hene, an inhomogeneity either in or in t On the other hand, if there is a wea inhomogeneity in and no transverse gradient (37), then virtually no refration of rays propagating at small angles ϑ ours Indeed, in this ase the dispersion equation is solved in the WKB approximation in the form (, x, ) for onstant, x, and ϑ tanϑ x / () Taing the form of the dispersion relations () into aount, we obtain ϑ() x ( + ξ() ), where ξ() Thus, the relative hange in ϑ will always be small A onsiderable hange in ϑ an be L ASTRONOMY LETTERS Vol No 4 996

7 488 aused by the transverse gradient In this ase, if there is no longitudinal gradient, ϑ x (x)/, and the relative hange in ϑ an be rather large, beause x varies near ero Thus, we may onsider the longitudinal density gradient to be responsible only for wave transformation, and the transverse gradient only for refration However, wave refration an produe a hange in the angle ϑ, whih determines wave transformation Therefore, in this setion we will onsider the evolution of ϑ during the propagation, assuming that no wave transformation ours, and then substitute these values of ϑ into the estimates for wave transformation obtained in setion 4 Let us now onsider a more general ase that taes the inhomogeneity of the magneti field into aount A magneti-field inhomogeneity produes the same effet of ray bending as ours in the presene of a transverse density gradient In general, the urvature of the ray path at a given point is determined by the omponent of the refrative-index gradient at this point that is perpendiular to We will use the Hamiltonian formalism to obtain a quantitative estimate of the ray urvature in this problem (see, eg, Kravtsov and Orlov 98) Let us onsider the propagation of waves and in approximations (7) and (9) adopted in setion, allowing us to use the approximate solutions () to the dispersion equation Sine waves in pulsars are generated mainly with frequenies (see setion ), our estimates may be onsidered to be valid for the propagation of waves from the site of their generation, orresponding to δ , to the transformation region, inlusive For the time being, we will ignore the effets of linear transformation, assuming that the waves are independent We define a Cartesian oordinate system x'o'' with its origin at the pulsar enter and the ' axis oiniding with the magneti axis In the subsequent analysis, we will use only this oordinate system and omit the primes for brevity Sine the waves propagate in a narrow region x near the pulsar magneti axis, we an write for the angle α between the unit vetor e Ç of the magneti field and the axis, assuming a dipole field 3 α -- x (38) -- Sine the urvature of magneti-field lines in the wavelength sale is very small, we will assume that the dispersion equation in the loal oordinates assoiated with the magneti field and used in setions 4 has the form () It an then be written in a form that is independent of the oordinate system by maing the substi BLIOKH, LYUBARSKII tution (, e Ç ), x [, e Ç ] --- ϑ Passing to the variables r and p -----, we obtain N δ() r, ( rp, ) ± 4γ δ() r [ pe, Ç () r ] γ 8γ (39) It is onvenient to write the Hamiltonians of the waves as H, ( p, r ) -- p ( N, ( rp, )) Vanishing of the Hamiltonian implies that the dispersion equation is fulfilled The equations for the rays have the form dr dτ H p, (4) p N s( pr, ) p dp H (4) dτ r N r In what follows, we omit the subsripts of the refrative indies for brevity The diretion of the vetor s oinides with that of the ray, ie, with the normal to the phase front of the wave, and its length relates the parameter τ to the ar length dσ s dτ Let us show that s Indeed, the first term in (4) p, beause the phase veloities of the waves are lose to, while the seond term is small: N p δ [ pe, Ç ] γ 8γ -- γ -- Thus, from (4) we approximately have dp dσ N( pr, ) r [ pe, Ç ] γ ASTRONOMY LETTERS Vol No (4) Hene, the hange in the angle β between p and the axis is dβ p, N( pr, ) dσ r (43) Note that the urvature of wave paths is determined only by the omponent of the refrative-index gradient that is perpendiular to, ie, in this problem by the N( pr, ) value of Substituting (39) into (43) gives the x

8 LINEAR TRANSFORMATION AND THE ESCAPE OF WAVES 489 following estimate for the hange of the angle between the axis and the wave vetor: dβ -- (44) γ d γ L x Here the ar element dσ is replaed by d, beause the wave propagates essentially along To solve the problem of linear transformation of waves, the evolution of the angle ϑ between the wave vetor and the magneti field is of interest For this purpose, let us determine the hange in the angle α between the unit vetor e Ç and the axis for a small shift along the ray, ie, along the vetor s Let the angle between s and e Ç at the point in question (x, ) be ϕ It follows from (4) and estimate (4) that the angle between p and e Ç is smaller than or of the order of -- ; γ sine the angle between p and e Ç ϑ --, we may onlude that ϕ ~ -- The small hange in the angle α is γ γ dα 3x along s ϕ d 4 ASTRONOMY LETTERS Vol No (45) Sine typial values for most of the pulsars are x/ ~ and γ ~, we will assume here that x/ γ In this ase, dβ dα, and the urvature of the ray path may be negleted ompared to the urvature of the magneti-field lines For the angle ϑ we have dϑ x d (46) In setion 3, we derived the ondition for strong wave interation (35), whih imposes a restrition on the angle ϑ Sine ϑ hanges in aordane with (46), for signifiant transformation we should require that the angle ϑ hange in the region of wave interation (36) in the range (35) We obtain a stronger ondition for ϑ: ϑ ---- γx λ ----, L (47) ie, the transformation will be signifiant only for waves that arrive at the region at angles satisfying (47) If we adopt typial pulsar parameters -- ~, x γ ~ λ, and ---- ~ 6, then the numerial value of the L harateristi angle (47) will be ϑ 4 As was mentioned above, waves in pulsars are mainly generated in the region /, orresponding to δ ~ This implies that, aording to (46), the hange in ϑ from the generation region to the transformation region is of the x order of -- Waves are predominantly generated at x angles ϑ -- --, and, therefore, they will ome to the γ x interation region at angles ϑ ~ -- ~ that are too large for the transformation to tae plae Thus, the more omplete analysis of wave propagation performed in this setion lends support to the onlusion reahed in setion 4: under onditions typial of pulsars, linear transformation of waves is not signifiant, beause the harateristi transformation angle (47) is small ompared to the harateristi angles of wave propagation in pulsar magnetospheres 6 ANALYSIS OF KINETIC EFFECTS In the above alulations, we assumed for simpliity that the plasma partiles all have the same momenta, orresponding to the Lorent fator γ In fat, this is not the ase In pulsar plasma, there is a wide spetrum of partile momenta, desribed by the momentum distribution funtion f(p) Sine in an infinitely strong magneti field eletrons and positrons affet the dispersion properties of waves in a similar way, f(p) should be taen as the sum of the distribution funtions of the eletrons and positrons It is ommonly assumed that the distribution funtion has a maximum in the region of momenta orresponding to the Lorent fator γ ~, whereas the harateristi minimum and maximum momenta of the partiles orrespond to γ min ~ and γ max ~ 3 respetively An important point is (see, eg, Lyubarsii 995) that the dispersion properties of eah wave are determined only by those partiles whose momenta are of the order of m p * ,, ie, only partiles of its own assoiated group are involved in the osillations of eah wave Aordingly, the properties of the wave will be the same as in a monoenergeti plasma of density f(p )p moving with * * the Lorent fator p /m Partiles with γ ~ γ * max orrespond to waves near the interation region, ie, near the point of intersetion of the dispersion urves Therefore, to estimate the frequeny, it is neessary to substitute γ max in plae of γ and f(p max )p max in plae of the total partile density n in formula () At the same time, stimulated sattering transfers the initial wave to the frequeny that is the plasma frequeny of the partiles at the pea of the distribution funtion In the laboratory referene frame, this frequeny is approximately equal to γ 4f( γ ) (Lyubarsii 995, 996)

9 49 BLIOKH, LYUBARSKII In a monoenergeti plasma, γ max γ and, whih was used above If, however, the spread of the partiles in momentum is great, the ratio / is inreased Therefore, a wave with frequeny must travel a longer path in the diretion of density derease to reah the region of possible transformation, where the loal value of is equal to the wave frequeny However, the longer the distane the wave travels in the magnetosphere, the greater the angle between the magneti field and the wave vetor beomes Thus, if the pulsar plasma shows a large spread of partiles in energy, the onditions for transformation an only deteriorate, and the onlusion that most of the waves do not undergo transformation and esape the pulsar magnetosphere in the form of transverse waves remains valid 7 CONCLUSION In this paper, we have onsidered the possibility of linear transformation of waves generated in pulsar magnetospheres to other types of waves The twostream instability generates mainly longitudinal waves, whih, due to indued sattering, are transferred to the region of superluminal phase veloities While propagating in the diretion of plasma-density derease, these waves beome transverse and eventually esape from the pulsar magnetosphere in the form of ordinary eletromagneti waves However, linear transformation may ause these waves to pass over to the other branh of the dispersion urve, where they remain nearly longitudinal and are finally absorbed due to Landau damping In this wor, we have obtained onditions for suh a transformation of waves We have shown that, under onditions typial of pulsars, the mehanism of wave interation onsidered here annot transform a signifiant fration of the generated waves into another type of wave Thus, the main result of this wor is that the linear transformation of waves in pulsar magnetospheres does not signifiantly impede the free esape of superluminal waves from the magnetospheres Therefore, the two-stream instability with subsequent stimulated sattering of the generated waves into the region of superluminal phase veloities may be a soure of pulsar radio emission ACKNOWLEDGMENTS This wor was partly supported by the Soros International Eduation Foundation in the Field of Exat Sienes (grant no GSU555) REFERENCES Arons, J and Barnard, JJ, Astrophys J, 986, vol 3, p Asseo, E, Pellat, R, and Rosado, M, Astrophys J, 98, vol 39, p 66 Asseo, E, Pellat, R, and Sol, H, Astrophys J, 983, vol 66, p Barnard, JJ and Arons, J, Astrophys J, 986, vol 3, p 38 Cheng, AF and Ruderman, MA, Astrophys J, 977, vol, p 8 Egorenov, VD, Lominade, DhG, and Mamrade, PG, Astrofiia, 983, vol 9, p 753 Ginburg, VL, Rasprostranenie eletromagnitnyh voln v plame (Propagation of Eletromagneti Waves in Plasma), Mosow: Naua, 967 Kravtsov, YuA and Orlov, YuI, Geometrihesaya optia neodnorodnyh sred (Ray Optis of Inhomogeneous Media), Mosow: Naua, 98 Lominade, DhG, Mihailovsii, AB, and Sagdeev, RZ, Zh Esp Teor Fi, 979, vol 77, p 95 Lominade, DhG, Mahabeli, GZ, Meliide, GI, and Pataraya, AD, Fi Plamy, 986, vol, p 33 Lyubarsii, YuE, Astron Lett, 993, vol 9, p 4 Lyubarsii, YuE, Astrophys Spae Phys Rev, 995, vol 9, p Lyubarsii, YuE, Astron Astrophys, 996 (in press) Ursov, VV and Usov VV, Astrophys Spae Si, 988, vol 4, p 35 Usov, VV, Astrophys J, 987, vol 3, p 333 Voloitin, AS, Krasnosel sih, VV, and Mahabeli, GZ, Fi Plamy, 985, vol, p 53 Weatherall, JC, Astrophys J, 994, vol 48, p 6 Zaslavsii, GM, Meitlis, VP, and Filoneno, NN, Vaimodeistvie voln v neodnorodnyh sredah (Wave Interation in Inhomogeneous Media), Novosibirs: Naua, 98 Zhelenyaov, VV, Eletromagnitnye volny v osmihesoi plame (Eletromagneti Waves in Cosmi Plasma), Mosow: Naua, 977 Zhelenyaov, VV, Koharovsii, VV, and Koharovsii, VlV, Usp Fi Nau, 983, vol 4, p 57 Translated by G Rudnitsii ASTRONOMY LETTERS Vol No 4 996

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