G. Z. Machabeli,a) S. V. Vladimirov, D. B. Melrose, and Q. Luo. Special Research Centre for Theoretical Astrophysics, School of Physics, The University of ...
PHYSICS OF PLASMAS
VOLUME 7, NUMBER 4
APRIL 2000
Particle acceleration by a fast ordinary mode in an electron–positron plasma G. Z. Machabeli,a) S. V. Vladimirov, D. B. Melrose, and Q. Luo Special Research Centre for Theoretical Astrophysics, School of Physics, The University of Sydney, New South Wales 2006, Australia
共Received 10 September 1999; accepted 22 December 1999兲 The possibility of nonresonant particle acceleration in an electron–positron plasma of a pulsar magnetosphere is investigated. A mechanism is proposed in which modulations of a fast superluminal 共with phase velocity exceeding the speed of light兲 longitudinal ordinary mode 共caused by a beat wave of two transverse electromagnetic waves propagating along the magnetic field兲 stimulate nonresonant quasilinear diffusion leading to a redistribution of plasma particles in pitch angle. The resulting perpendicular momenta of the particles lead to synchrotron radiation which is in the ␥-ray range. © 2000 American Institute of Physics. 关S1070-664X共00兲01304-5兴
I. INTRODUCTION
synchrotron radiation, which produces the high-energy radiation. Thus we are concerned with the possibility that nonlinear processes allow the energy stored in superluminal Langmuir waves to be finally converted into very high energy photons through this sequence of processes. To describe nonlinear wave processes in a pulsar magnetosphere, it is not sufficient to use only the approximation of weak collective fields 兩 E 兩 2 Ⰶmc 2 n ␥ , where m is the electron 共positron兲 mass, n⭐1015⫺1016 cm⫺3 is the number density of the main plasma particles, and ␥ ⫽1/冑1⫺ v 2 /c 2 is the Lorentz factor; it is also necessary to take into account the 共small兲 parameter of the drift approximation ( pe / B )Ⰶ1, where pe ⫽(4 ne 2 /m) 1/2 is the electron plasma frequency, B ⫽eB 0 /mc is the cyclotron frequency, and B 0 ⯝1012 G is the pulsar’s magnetic field near the surface of the neutron star. The magnetic field, assumed to be dipolar, decreases away from the star proportional to the cube of the distance. Thus the parameter p / B increases with the increasing distance, but it is much less than unity in the whole magnetospheric region confined by the light cylinder 共the latter is the surface on which the velocity of particles corotating with the magnetic field is equal to the speed of light兲. We are concerned with nonlinear processes for slightly oblique waves 关i.e., waves propagating with small angle ⍜ ⬃k⬜ /k z with respect to the magnetic field B0 ⫽(0,0,B 0 )]. In the reference frame of the observer, this approximation is quite general since the plasma of interest moves toward the observer with Lorentz factor ␥ p Ⰷ1, and the angle of the observed radiation can be estimated as ⍜⭐1/2␥ p . It is well known that the decay l→l ⬘ ⫹l ⬙ of a Langmuir wave into two Langmuir waves is forbidden in such a plasma; it was demonstrated6 that the decay l→2t into two transverse waves is possible 共as well as the reverse process; the latter was applied in Ref. 7 to explain the pulsar radio emission兲. Three wave decay processes and calculation of the wave spectra were investigated in Ref. 8, and nonlinear scattering of Langmuir waves on plasma particles 共applied to pulsar magnetosphere兲 was considered in Ref. 9. Analysis of these processes shows that in pulsar magnetospheres there is ex-
The most developed models of a pulsar magnetosphere1–3 are based on the idea proposed in Ref. 4 and the hypothesis of Ref. 5 that the plasma is formed by electrons and positrons. The e ⫺ e ⫹ pairs are created by ␥ quanta in the strong external magnetic field B0 . The perpendicular 共with respect to the field B0 ) component of the momentum of the created particles rapidly disappears due to synchrotron losses, and the particles move in one dimension along the magnetic field lines. Notable features of the pulsar magnetospheric plasma are the absence of gyrotropy 共because the masses of electrons and positrons are equal兲, the relativistic character of particle motion, and the one-dimensional particle distribution function. The plasma particles move with relativistic velocities along the external magnetic field lines and can generate the natural modes of such a plasma. It is thought that pulsar radio emission results from the relativistic particles exciting the natural wave modes of the plasma, with some mechanism allowing conversion into escaping radiation. Thus most of the properties of the radio emission are formed in the pulsar magnetosphere. Naturally, in the magnetospheric plasma the electromagnetic modes generated include not only those that can leave the magnetosphere, but also include Langmuir waves. In this paper we are concerned with nonlinear effects associated with superluminal 共phase speed v ph ⬎c) Langmuir modes that can result from nonlinear wave–wave interactions in the plasma. These waves experience no Landau damping, so with even a modest generation mechanism their energy density can easily build up to a level where nonlinear processes become possible. Of particular interest for us here is the possibility that the nonlinear processes lead to pitch-angle scattering of the relativistic particles. If the pitch angle is such that the perpendicular momentum of the particles greater exceeds mc 共where m is their mass兲, then they radiate a兲
Permanent address: Abastumani Astrophysical Observatory, 2a, A. Kazbegi, Tbilisi 380060, Georgia.
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Particle acceleration by a fast ordinary mode in an . . .
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cessive energy stored in the superluminal Langmuir waves. Conversion of the wave energy into energy of plasma particles and possible consequences of the process is the subject of the present investigation. The paper is organized as follows: In Sec. II, we briefly review the main results of the linear theory of collective wave and particle processes in a relativistic electron– positron plasma; the motion of a test particle is investigated in Sec. III; Sec. IV is devoted to a study of the dynamics of the nonlinear waves, and in Sec. V we consider particle acceleration and the possibility of increasing pitch-angles because of the nonresonant interaction, resulting in synchrotron radiation. Application of the results to the observed features of the pulsar ␥-emission is discussed in Sec. VI. II. WAVES IN PULSAR MAGNETOSPHERES
The curvature of the pulsar magnetic field can be neglected for wave propagation at wave lengths which satisfy the inequality B Ⰶ1, B r
共1兲
i.e., when the wave length is much less than the scale length of the inhomogeneity of the magnetic field. We may then introduce Cartesian coordinates with the z-axis directed along the magnetic field, and the x-axis directed along the perpendicular component k⬜ of the wave vector k. We assume the plasma being cold and nonrelativistic in its rest frame. The dispersion equation factorizes into two independent equations. One describes a purely transverse electromagnetic mode with the dispersion10
冉
x ⫽kc 1⫺
冊
1 2p 1 , 8 B2 ␥ 30
共2兲
where p ⫽ 关 4 (n e ⫹n e⫹ )e 2 /m 兴 1/2⫽ 冑2 pe is the ‘‘combined’’ plasma frequency 共i.e., taking into account contributions of plasma electrons and positrons兲, and ␥ 0 is the Lorentz factor of plasma particles moving with velocity v 0 along the field lines. Below, we consider a strongly magnetized ˜ B , where ˜ B⫽ B / ␥ 0 . relativistic plasma, p Ⰶ The second factor of the dispersion equation describes two modes of mixed longitudinal-transverse character. The lower-frequency subluminal mode is analogous to the Alfve´n wave, and the higher-frequency mode which can be superluminal, v ph ⬎c, is the ordinary o-mode. Expressions for the dispersion of these modes can be easily written in some limits. We consider the case kcⰆ p for waves propagating almost parallel to the magnetic field 兩 k⬜ 兩 Ⰶk z . Thus for the o-mode we have
2o ⯝
2p
␥ 30
FIG. 1. Spectra of waves in an electron-positron plasma for oblique propagation.
⫹3k z2 c 2 ⫹ 兩 k⬜ 兩 2 c 2 .
ties must take this into account. The electric field of the x-mode is perpendicular to the plane of vectors k and B; the electric fields of the o-mode and Alfve´n mode are in the plane. For frequencies close to the plasma frequency p : the latter modes are almost longitudinal, and for frequencies far from p , both modes are almost transverse. Low-frequency modes analogous to the ion-acoustic wave in an electron-ion plasma, are absent in an electronpositron plasma. Thus when considering nonlinear effects in the wave propagation, the only possibility for amplitude modulations of the o-mode is due to nonresonant excitation of a beat wave. This was first pointed out in Ref. 11, see also Refs. 12–14, and considered in detail in Ref. 15, where the possibility of modulations of the superluminal o-mode by
共3兲
Graphically, the dispersion relations 共2兲 and 共3兲 are presented in Fig. 1. For parallel propagation there is also a point p ⬇k 0 c, see Fig. 2, where the dispersion relations for all three modes coincide and proper consideration of their nonlinear proper-
FIG. 2. Spectra of waves in an electron-positron plasma for parallel propagation.
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transverse waves investigated 共by ‘‘transverse’’ in Ref. 15 implies not only the x-mode, but also the higher-frequency, compared with p , o-mode, when its dispersion is close to the vacuum case兲. In this interaction, the longitudinal superluminal component of the perturbation appears as a result of the interaction of two transverse waves with
冏
冏
t⫺ ⬘t ⬎c. k z ⫺k z⬘
共4兲
Substituting 共2兲 into 共4兲, and using 兩 k兩 ⯝k z (1⫺k⬜2 /2k z2 ), this inequality implies
冋
册
k⬜2 k⬜⬘ 2 1 1⫺ ⫺ ⬎1. 2 k z 共 k z ⫺k z⬘ 兲 k z⬘ 共 k z ⫺k z⬘ 兲
共5兲
To satisfy 共5兲 for k z ⬎k z⬘ , it is required that 兩 k⬜⬘ 兩 ⬎ 兩 k⬜ 兩 as well as 兩 k⬜⬘ 兩 兩 k⬜⬘ 兩 ⫺ 兩 k⬜ 兩
kz
k z ⫺k z⬘
⬎1.
共6兲
d p 1x ⫽e 共 E x ⫹E x⬘ 兲 ⫹ ˜ B 共 p 1y cos ˜ B ⫹ p 1x sin ˜ B 兲 cos ˜B, dt d p 1y ⫽e 共 E y ⫹E ⬘y 兲 ⫺ ˜ B 共 p 1x sin ˜ B ⫹ p 1y cos ˜ B 兲 sin ˜B. dt
共13兲
p 1x ⫽
e 兩 E⬜ 兩 sin t e 兩 E⬜⬘ 兩 sin ⬘ t ⫹ , 2共 ⫹ ˜ B兲 2共 ⬘⫹ ˜ B兲
p 1x ⫽⫺
e 兩 E⬜ 兩 sin t e 兩 E⬜⬘ 兩 sin ⬘ t ⫹ . 2共 ⫹ ˜ B兲 2共 ⬘⫹ ˜ B兲
共14兲 共15兲
We also assume that E y ⫽ 兩 E⬜ 共 r,t 兲 兩 sin t,
共16兲
as well as
The equation of motion for a test particle moving together with the plasma 共i.e., in our reference frame p 0z ⫽0) in the external magnetic field and the fields of the incident and scattered waves is
冋
共12兲
The solution of 共12兲 and 共13兲 is given by
E x ⫽ 兩 E⬜ 共 r,t 兲 兩 cos t,
III. MOTION OF PARTICLES
册
1 dp ⫽e E⫹E⬘ ⫹ v⫻ 共 B0 ⫹B⫹B⬘ 兲 . dt c
共7兲
There are two small parameters in the problem. First, there is the smallness of the wave fields as compared to the external magnetic field, (E,E⬘ ,B,B⬘ )ⰆB0 . Second, there is the smallness of the wave energies compared to the plasma particle thermal energy, ( 兩 E兩 2 , 兩 E⬘ 兩 2 )Ⰶmc 2 n ␥ T . Furthermore, we split the test particle momentum into three parts, p⫽p0⬜ ⫹p1 ⫹p2 ,
共8兲
where p0⬜ corresponds to the unperturbed motion of the particle in the external magnetic field B0 , p1 Ⰶp0⬜ is the linear perturbation of p0⬜ due to the waves, and p2 Ⰶp1 is the nonlinear perturbation of the particle motion. Expansion of the Lorentz-factor ␥ ⫽ 关 1⫹(p0⬜ ⫹p1 ) 2 /m 2 c 2 兴 1/2 to first order gives p1 p0⬜ v 共 p0 •p1 兲 p0 ⫹ ⫺ , ⯝ 3 c mc ␥ 0⬜ mc ␥ 0⬜ mc ␥ 0⬜
共9兲
2 /m 2 c 2 兴 1/2. After substitution of this exwhere ␥ 0⬜ ⫽ 关 1⫹p0⬜ pression into Eq. 共7兲, which is expanded in the same way, we find a system of coupled equations for p0⬜ , p1 , and p2 . In the zeroth approximation, we have
dp0⬜ e ⫽ p ⫻B0 . dt mc 0⬜
共10兲
If the test particle is an electron, the solution of 共10兲 is p 0x ⫽p 0⬜ cos ˜ B t,
In the first approximation, the equation of motion can be written in the form
p 0y ⫽⫺p 0⬜ sin ˜ B t,
共11兲
˜ B ⫽ B / ␥ 0⬜ . For a positron, the sign of the where y-component of the momentum is opposite.
E x⬘ ⫽ 兩 E⬜⬘ 共 r,t 兲 兩 cos ⬘ t,
E ⬘y ⫽⫺ 兩 E⬜⬘ 共 r,t 兲 兩 sin ⬘ t,
共17兲
where the wave amplitudes E⬜( ⬘ ) (r,t) are slow functions of position and time. For positrons, we have similar solutions with the change e→⫺e and therefore B →⫺ B as well. Taking into account that the parallel component of p2 ⫻B0 is zero, we have e d p 2z ⫽ 关 p ⫻ 共 B⫹B⬘ 兲兴 z . dt mc ␥ 0⬜ 1
共18兲
From Maxwell’s equations, we have B x⫽
Ey , cos ⍜
B y ⫽⫺
Ex , cos ⍜
共19兲
where we use the wave dispersion equation ⯝ 兩 k兩 c and introduce the angle ⍜ between the external magnetic field and the wave vector: cos ⍜⫽kz /兩k兩 . Substituting 共14兲, 共15兲, and 共19兲 into Eq. 共18兲, we obtain the nonlinear longitudinal perturbation equation
冋
册
p 1 p 1 d p 2z e 2 E⬜2 ⫽ ⫺ sin共 ⌬ t 兲 , dt mc p ˜ B ⫹ cos ⍜ ˜ B ⫹ ⬘ cos ⍜ ⬘ 共20兲 t t where we introduce ⌬ ⫽ ⫺ ⬘ . The electric field E 2z is the result of the nonlinear interaction of the waves t and t ⬘ with the plasma particles. ˜ B 共note we have p Ⰶ ˜ B ). In Consider the limit t Ⰷ this case plasma particles do not have time to complete one Larmor cycle, and generation of the wave t ⬘ is due to reradiation of the wave t by the particle whose unperturbed motion is effectively rectilinear. For simplicity, we do not consider the possibility of generation on higher cyclotron harmonics. In this case, the frequency of the radiated wave ⬘ t is close to the frequency of the incident wave t . For cos ⍜⯝cos ⍜⬘⬇1 and 兩 E⬜ 兩 ⬇ 兩 E⬜⬘ 兩 , from Eq. 共20兲 we obtain in the dimensionless variables
Phys. Plasmas, Vol. 7, No. 4, April 2000
E→
eE , mc p
0r r→ * , c
Particle acceleration by a fast ordinary mode in an . . .
t→
*
0 t,
p→p/mc 共21兲
dp 2 ⫽ dt
0 ˜ 兩 Et 兩 2 *2 B ⬜ ␥ 0⬜
sin共 ⌬ t 兲 ,
共22兲
3 where 0 ⫽ p / ␥ 0⬜ . * For further convenience, we rewrite Eq. 共22兲 as
dp 2 ⫽a sin共 ⌬ t 兲 , dt
*0
,
0 ˜ B 兩 E⬜t 兩 2 . a⫽ * 2 ␥ 0⬜
共24兲
From Eq. 共23兲, we have the solution a 共 1⫺cos ⌬ t 兲 . ⌬
共25兲
IV. NONLINEAR DYNAMICS OF THE O-MODE
Detailed investigation of the instability leading to the nonlinear modulation of the o-mode by two high-frequency transverse waves was done in Ref. 15, where we refer the interested reader. Here, we briefly remind the reader of the main results of the study. Force 共22兲 demonstrates nonlinear coupling of longitudinal components E l with transverse components E t . To describe the nonlinear dynamics, we start from Maxwell equations and consider a wave packet propagating at a small angle with respect to the external magnetic field. Separating the low-frequency and high-frequency transverse components, E⫽El ⫹Et , where E zt ⫽0, we assume t Ⰷ l . Introducing slowly changing wave amplitudes and density perturbations ␦ n, we obtain
E 共t x,y 兲 ⫽0, t
共26兲
and l k l0 c E zl l E 共 x,y 兲 ␦n l ⫺i ⫽ E , 2i 0 t 0 共 x,y 兲 n 0 共 x,y 兲 * * 共27兲 k l0 c E lx E ly l E zl ␦n l ⫺i ⫹ ⫽ E . 2i 0 t 0 x y n0 z * * Equations 共26兲 are written in the zeroth approximation for the expansion in the parameter l / t . The nonlinear terms on the right-hand sides of Eqs. 共27兲 are determined by the amplitudes of the high-frequency waves E⬜t . The expression for ␦ n/n 0 is found by averaging the continuity equation over the high frequency
冉
␦n ⫽ p 2z . t n0 z
Thus we obtain
冊
共28兲
Note that ␦ n/ x⫽ ␦ n/ y⫽0, and p 2z is defined by 共25兲.
共29兲
冋冕 冉 冊 册
共 ⵜ⫻El 兲 z ⫽C exp
共23兲
where ⌬ ⫽( t ⫺ ⬘ t )/
p 2z 共 t 兲 ⫽
Excluding the term with E zl , we find from the first equation of the system 共27兲
␦n 共 ⵜ⫻El 兲 z ⫽ 共 ⵜ⫻El 兲 z . t n0
the following equation:
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t
␦n n0
dt ,
共30兲
where C is a constant. Since there is an instability of the curl field, below we assume that ⵜ⬜ •E⬜l ⫽0. Thus in the presence of the density perturbation there is exponential growth of the transverse fields. We note that in the drift approximation the density modulation as well as the change of the momentum p 2 , caused by the high-frequency fields E t 共when k储 B0 ), are parallel to the axis z储 B0 . However, the growth of the fields due to the parallel density modulations may be in any direcl ⬀exp(⫺ikzz⫺ik⬜ •r⬜ ). tion: E (x,y,z) To obtain an equation for the parallel component E zl , we l find the mixed derivatives 2 E (x,y) / t (x,y) from the first equation of 共27兲. Thus differentiating the second equation of 共27兲 with respect to time 共i.e., applying / t), substituting the resulting expression for the mixed derivatives, and assuming ⵜ⬜ •E⬜l ⫽0 共which implies that there are no components of the potential electric field perpendicular to the external magnetic field兲, we obtain
冉 冊
2 E zl i ␦n l ⫺K 20 ⌬⬜ E zl ⫹ E ⫽0, t2 2 t n0 z
共31兲
where K 0 ⫽k l0 c/2 0 . The nonlinear term is proportional to * (E⬜t ) 2 E zl , in our case, we have (E⬜t ) 2 E zl Ⰷ 兩 E zl 兩 3 . The dispersion equation is given by
2 ⫺K 20 k⬜2 ⫹
k za ⫽0. ⌬
共32兲
Thus the instability is possible when k za ⬎K 20 k⬜2 . ⌬
共33兲
In the approximation considered, the aperiodic growth of the longitudinal potential field is not accompanied by a density modulation since the latter, as the continuity equation 共28兲 implies, is determined by the high-frequency transverse fields. The energy of the high-frequency t-modes is assumed to be maintained by external sources, which is reasonable for the plasma in a pulsar magnetosphere where excitation of the transverse modes should be very effective.16–18 From Eq. 共28兲, we also find that
␦n n0
冕
⫽⫺i
k za ⌬
⫽⫺i
k za 1 t⫺ sin ⌬ t . ⌬ ⌬
冉
t
0
共 1⫺cos ⌬ t 兲 dt ⬘
冊
共34兲
Substituting this equation into 共30兲 and using the expansion ⫹⬁
exp共 i ␣ sin ⌬ t 兲 ⫽
兺
s⫽⫺⬁
J s 共 ␣ 兲 e is⌬ t ,
共35兲
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⫹⬁
共 ⵜ⫻El 兲 z ⫽C
兺 Jn s⫽⫺⬁
冉
k za 共 ⌬ 兲2
冊 冋 冉 exp ⫺i
k za 共 ⌬ 兲2
冊 册
⫺s ⌬ t .
⫹
共36兲
k za . 共 ⌬ 兲2
共37兲
In this case, (ⵜ⫻El ) z is determined by the Bessel function with equal index and argument, i.e., J ( )⬃ ⫺1/3. 19
V. PARTICLE ACCELERATION
According to the model,4 a rotating neutron star with corotating magnetic field generates an electric field with a component along B0 . The electric field ‘‘pulls’’ primary particles out of the surface of the star and accelerates them up to relativistic velocity. In the curved magnetic field of the pulsar magnetosphere, the relativistic particles emit ␥-quanta. When the energy of the ␥-quanta exceeds 2mc 2 , pair production starts, ␥ ⫹B→e⫹e ⫹ ⫹B, and populates the polarcap regions with secondary pairs. The region where the pair production takes place 共and the electric field is effectively screened兲, is called the pair production front 共PPF兲.3 The secondary pair plasma moves along the magnetic field lines with relativistic velocity; the primary relativistic beam moves through it. It is assumed that the energy density of the secondary plasma and the primary beam are comparable, n p ␥ p ⬇n b ␥ b . The beam density is of the order of the Goldreich–Julian density n b ⬃n GJ ⬇1012 cm⫺3 , and ␥ b ⯝3 ⫻106 . As demonstrated above, the instability of superluminal Langmuir waves can develop on density modulations induced by the beat of two transverse waves. The developed modulational perturbation is of ‘‘cigar’’-like type, 兩 k⬜ ⫺k⬜⬘ 兩 ⬎ 兩 k z ⫺k z⬘ 兩 . Thus the energy of the Langmuir waves is converted into low-frequency perturbations ⫺⬘. The ponderomotive force acts on plasma particles in the parallel as well as in the perpendicular dierections, and there is nonresonant interaction of the unstable fields with plasma particles. To describe the interaction, we write the nonresonant quasilinear equation20 which, e.g., was used in Ref. 10 for the relativistic case in the approximation k⬜ p⬜ Ⰶ1. mB We have 共back in dimensional variables兲
共38兲
冉 冉
k
e2 f 兩 E kz 兩 2 ␦ 共 k ⫺k z v z 兲 2 pz
冊
冉
册
E⬜2 p z2 f B⬜2 p ⫹ t 4B 20 p⬜ p⬜ ⬜ p⬜ t 4B 20
⫻ p⬜2
From the latter expression, we see that 共averaging over sufficiently long time interval TⰇ1/⌬ ) the transverse perturbation generated by the density modulation is not zero when
s⫽
冋兺
f ⫽ t pz
we obtain
冊
冊
2 f p z2 f f p⬜ ⫺2 p⬜ pz , 2⫹ p z p⬜ p z p⬜ p⬜ p⬜ 共39兲
⬘ 兩 2 ⫹ 兩 E ⬘ky 兩 2 ) ⬘ 兩2 and B⬜2 ⫽ 兺 k ( 兩 B kx where E⬜2 ⫽ 兺 k ( 兩 E kx 2 ⫹ 兩 B ⬘ky 兩 ). We should remark on the assumption that p⬜ may be treated as a continuous variable when, for some other purposes, the quantization of the perpendicular motion in the strong magnetic field needs to be taken into account. The quantization involves discrete values of p⬜ separated by ⬃mc(B/B c ), with B c ⫽4⫻1013 G. We are interested in the case p⬜ Ⰷmc, when synchrotron radiation is possible, and then for (B/B c )Ⰶ1 the quantization of the energy levels is unimportant and p⬜ may be regarded as a continuous variable. Because the initial distribution function is almost onedimensional, p⬜ Ⰶ p 储 , we assume that in 共39兲 ( / p⬜ ) Ⰷ( / p z ). Taking into account rapid change of the distribution function, assuming the magnetic fluctuations to be dominant, and keeping the largest contribution to the averaged 共over momenta, see below兲 quantities, we obtain
冉
f B⬜2 ⬇ t t 4B 20
冊冉
p⬜2
冊
2 f p z2 f f p⬜ ⫺2 p⬜ pz . 2⫹ p z p⬜ p z p⬜ p⬜ p⬜ 共40兲
Next, we multiply 共40兲 by p⬜2 and integrate over momenta, and multiply 共40兲 by p z2 and integrate over momenta to find
冉 冊
具 p⬜2 共 t 兲 典 B⬜2 ⬇6 具 p z2 共 t 兲 典 t t 4B 20 and
冉 冊
具 p z2 共 t 兲 典 B⬜2 2 ⬇⫺8 具 p z 共 t 兲 典 , t t B 20
共41兲
共42兲
where 具...典 means averaging over particle momenta. Substituting
具 p⬜2 共 t 兲 典 具 p⬜2 共 t 兲 典 ⫽ t t 具 p z2 共 t 兲 典 具 p⬜2 共 t 兲 典 1
具 p⬜2 共 t 兲 典 1 具 p z2 共 t 兲 典 ⫹ 2 t 具 p z 共 t 兲 典 具 p z2 共 t 兲 典
共43兲
and using Eq. 共42兲, we obtain the expression for the pitchangle, assumed small,
⬇
冉
具 p⬜2 共 t 兲 典 具 p z2 共 t 兲 典
冊 冑冉 冊 1/2
⫽
3 B⬜ . 2 B0
We also have from Eq. 共42兲,
共44兲
Phys. Plasmas, Vol. 7, No. 4, April 2000
2 具 p z2 典 ⫽ 具 p z0 典 exp
冉
⫺
8B⬜2 B 20
冊
,
Particle acceleration by a fast ordinary mode in an . . .
共45兲
where p z0 is the parallel particle momentum at the initial moment, i.e., when B⬜ ⫽0. Since B⬜ ⰆB 0 everywhere in the magnetosphere, there is no significant change of the average parallel energy. This is expected because the nonresonant interaction redistributed the wave energy amongst all particles, giving relatively little energy to any single particle. The resonant wave–particle interaction is described by the first term on the right-hand side of Eq. 共39兲. This term is not zero only when the resonant condition k ⫽k z v z is satisfied, and this is not possible using the assumptions made here. Thus we find that the nonlinearly generated wave modulations lead to increased pitch-angles. As a particle acquires pitch-angle, it emits synchrotron radiation. Synchrotron radiation is accompanied by a radiation reaction force which can then be taken into account in the kinetic equation 共39兲, similarly to the resonant case.16,17 However, the contribution of the back reaction to the change of the pitch-angle can be neglected if the relevant plasma instability grows sufficiently fast. The radiation reaction due to synchrotron radiation can be characterized by the energy loss time of a single particle t s ⬇mc 2 ␥ ⬘ / P syn , where21 P syn⫽(1/4 ) T ( ␥ ⬘ ) 2 B 20 , T is the Thomson cross section, and ␥⬘ is the Lorentz factor of radiating particles in the plasma rest frame. This should be compared to the characteristic time scale t⬜ ⬃ 2 t z of the energy change associated with the particle’s perpendicular motion, cf. Eq. 共41兲: t s ⬎ 2 t z . To estimate t z , we note that in the equilibrium the parallel diffusion time should equal the growth rate of the relevant plasma instability, i.e., the rate of the energy transfer from the Langmuir condensate to the particles: t z ⬃(⌬ /k z a) 1/2. However, the coefficient a defined by 共24兲 can also depend on the synchrotron reaction force, thus making the full self-consistent treatment a very complex problem which is beyond the scope of the present paper. Here, we are interested in the possibility of ␥-ray emission due to nonresonant quasilinear diffusion and therefore it is reasonable to assume that t z should not be longer than the flow time of the plasma, R LC /c, where R LC is the lightcylinder adius, which is typically less than 1 s. Therefore, given the very small pitch angle ⭐10⫺5 and t z ⭐1 s, the condition t s ⬎t⬜ can easily be satisfied. For further estimations, we assume that the wave energy density is of the order of the beam energy density, viz. B⬜2 ⬃mc 2 n b ␥ b . We stress here that the amplitude of the growing transverse magnetic field perturbations B⬜ has the characteristic frequency ⌬ which is much less than the plasma frequency p . The pulsar magnetic field changes with distance according to B 0 ⫽B 0p
冉 冊 R0 R
,
共46兲
,
共47兲
also, we have n b ⫽n 0b
冉 冊 R0 R
where R 0 is the radius of the neutron star, B 0p ⬃1012 G, and n 0b ⬃1012 cm⫺3 . For young, rapidly rotating pulsars 共such as Crab and Vela兲, the radius of the light cylinder is R LC ⬃108 cm. The estimations show that in the rest frame,
⬇
3
冉
mc 2 n ⬘b ␥ b⬘ B 20
冊
1/2
,
共48兲
where n b⬘ ⬇n b /2␥ b and ␥ b⬘ ⬇ ␥ bT are the density and Lorentzfactor of the beam in the rest frame of the beam. Assuming that the width of the beam distribution function ␥ T is of the same order as ␥ b , for ␥ b ⬃ ␥ T ⬃107 , n b ⬃1012 cm⫺3 , B 0 ⬃106 G 共corresponding to R⯝102 R 0 ), and R 0 ⬃106 cm, we find that the pitch-angle can be of order 10⫺6 , which is large enough for the particles to emit significant synchrotron radiation. The synchrotron radiation in the rest frame of the beam is given by21
0 ⬇ B ␥ 2b ⬃1021 s⫺1 .
共49兲
In the reference frame of the observer, the frequency is
⫽
0 1 , ␥ 1⫺ v cos ⍜/c
共50兲
where ⍜ is the angle between k 共in our case this direction is to the observer兲 and B0 . For ⍜⬇0 we have ⯝2 0 ␥ b , which corresponds to 10 MeV. Thus the radiation can be in ␥-range for some pulsars and the considered process is a potentially important mechanism for the pulsar gamma-ray emission. Detailed application is presented elsewhere.22 VI. CONCLUSION
To conclude, we argue that t-modes generated in a pulsar magnetosphere can create beat density modulations along the magnetic field. When the modulation frequency ⌬ is much less than the frequency of the generated field perturbations, the growth of the parallel potential field E zl is accompanied by the growth of the transverse electromagnetic field E⬜ according to Eq. 共30兲. The energy of the modulations can then be converted into perpendicular momentum of the nonresonant particles and this in turn leads to the synchrotron radiation at high photon energies. Direct application of this mechanism to observational data includes case by case analysis of concrete results and is the subject for further investigations. ACKNOWLEDGMENT
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