# Propagation effects on polarization of pulsar radio emission

## Transcript Of Propagation effects on polarization of pulsar radio emission

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Mon. Not. R. Astron. Soc. 307, 830±840 (1999)

Propagation effects on polarization of pulsar radio emission

R. T. Gangadhara,1,2 H. Lesch3 and V. Krishan1

1Indian Institute of Astrophysics, Bangalore 560034, India 2National Centre for Radio Astrophysics, TIFR, Pune University Campus, Pune 411007, India 3Institut fuÈr Astronomie und Astrophysik der UniversitaÈt MuÈnchen, Scheinerstrasse 1, 81679 MuÈnchen, Germany

Accepted 1999 March 19. Received 1999 March 19; in original form 1998 May 6

ABSTRACT We consider the role of a propagation effect such as stimulated Raman scattering on the polarization of radio pulses. When an intense electromagnetic wave with frequency close to the plasma frequency interacts with the plasma in the pulsar magnetosphere, the incident wave undergoes stimulated Raman scattering. Using typical plasma and magnetic field parameters, we compute the growth rate and estimate the polarization properties of the scattered mode. Under some conditions, we find that the polarization properties of the scattered mode can become significantly different from those of the incident wave. The frequencies at which strong Raman scattering occurs in the outer parts of the magnetosphere fall into the observable radio band. In some pulsars, for example PSR B0628228 and B1914113, a large and symmetric type of circular polarization has been observed. We propose that such an unusual circular polarization is produced by the propagation effects.

Key words: plasmas ± polarization ± radiative transfer ± waves ± stars: magnetic fields ± pulsars: general.

1 INTRODUCTION

The investigation of coherent radio emission from pulsar magnetospheres has attracted a great deal of attention (Cordes 1979; Michel 1982; AsseÂo, Pelletier & Sol 1990). Powerful collective emission occurs when relativistic electron beams with density ,1 per cent of the pair plasma density scatter coherently from concentrations of plasma waves (cavitons) (Benford 1992). The role of collective plasma processes in the absorption and spectral modification of radio waves is well known (e.g. Beal 1990; Krishan & Wiita 1990; Benford 1992; Gangadhara & Krishan 1992, 1993, 1995; Gangadhara, Krishan & Shukla 1993). von Hoensbroech, Lesch and Kunzl (1998) have demonstrated that degree of linear polarization decreases with increasing frequency while the degree of circular polarization shows the opposite trend.

In this paper, we estimate the role of stimulated Raman scattering in the polarization of electromagnetic waves propagating in the pulsar magnetosphere. We assume that the physical conditions in the pulsar magnetosphere are those of the classical standard model (Ruderman & Sutherland 1975) which describes the generation of ultrarelativistic beams of electrons/positrons and the creation of the pair plasma. The beams and the pair plasma are in relativistic motion along the bundle of open magnetic field lines that delimit the active region of the magnetosphere. The stimulated Raman scattering is considered as a parametric decay of the initial transverse electromagnetic (pump) wave into another electromagnetic wave and a longitudinal plasma wave. The physics of stimulated Raman scattering in a plasma has been explained in many papers and books (e.g. Drake et al. 1974; Liu & Kaw 1976; Hasegawa 1978; Kruer 1988). There are two ways in which stimulated Raman scattering may be important in the pulsar environment. First, it may act as an effective damping mechanism for the electromagnetic waves generated by some emission mechanism at the lower altitudes in the pulsar magnetosphere. At those altitudes, the resonant conditions for stimulated Raman scattering, i.e. frequency and wavenumber matching, might not be satisfied. This results in a short time-scale of variability which is generally observed in pulsar radio emission. Secondly, stimulated Raman scattering may provide an effective saturation mechanism for the growth of the electromagnetic waves, provided that the conditions for wave excitation by some mechanism are satisfied in the region where effective stimulated Raman scattering can take place.

The first case can be simplified by treating the intensity of the pump as constant in time. Then the non-linear equations, which describe the wave coupling, become linear in the amplitudes of the decaying waves, and the exponentially growing solutions imply an effective energy transfer from the pump wave. This approximation breaks down when the amplitudes of the decay waves become comparable to that of the pump wave or when the amplitudes of the decay waves enter the non-linear stage, and start losing energy owing to some non-linear processes such as wave±particle trapping and acceleration.

The second case is more complicated, where stimulated Raman scattering acts as a non-linear saturation mechanism and the

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Polarization of pulsar radio emission 831

amplitudes of all waves may be of the same order. This case can be considerably simplified when damping of the plasma wave is very strong or if it leaves the region of the resonant interaction quickly enough.

We neglect the non-linear stages of stimulated Raman scattering and the development of Langmuir turbulence, which leads to waveparticle trapping or quasi-linear diffusion. If the pump is monochromatic, the growth rate of stimulated Raman scattering can become very high, as in conventional laboratory laser±plasma interaction. However, in the case of pulsars the pump can be broadband, and, in the limit where the bandwidth Dv of the pump wave is much larger than the growth rate of stimulated Raman scattering, we can use a random phase approximation for the statistical description of the interacting waves.

Tsytovich & Shvartsburg (1966) have given a general expression for the third-order non-linear currents excited in a magnetized plasma. Since the corresponding expressions are very complicated, the general case of Raman scattering becomes very difficult to consider. However, one can make some useful simplifications when considering stimulated Raman scattering in the pulsar magnetosphere. First, in a superstrong magnetic field, we can expand the currents in terms of 1/v B, where vB is the cyclotron frequency. Secondly, if the pair plasma has the same distributions for electrons and positrons then some of the non-linear currents cancel out, as they are proportional to the third power of the electric charge (this cancellation is exact in the unmagnetized electron±positron plasma). Thirdly, all the three interacting waves propagate along the magnetic field. This is an important but less justified approximation. It allows us to simplify the problem considerably, and to obtain a dispersion relation for stimulated Raman scattering.

The polarization of the pulsar signals appear to bear critically on the pulsar radio emission process and the emission beam geometry. One or more reversals of the sense of circular polarization have been observed in the intergrated profiles of several pulsars. However, in individual pulses, circular polarization changes sense many times across the pulse window. Further, it is important to determine whether the depolarization is a geometric effect or results from radiation±plasma interactions. There have only been very preliminary attempts to explain depolarization and microvariability using plasma mechanisms (Benford 1992).

Our purpose in this paper is to derive the properties of natural plasma modes and to explore some possible implications concerning the interpretation of the observed polarization, notably large and symmetric circular polarization in some pulsars (e.g. PSR B0628±28 and B1914113). In Section 2, we derive the dispersion relation for stimulated Raman scattering of an electromagnetic wave, and give an analytical expression for the growth rate of the instability. In Section 3, we define the Stokes parameters and compute the polarization states of the scattered electromagnetic wave. The discussion of our findings is given in Section 4.

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2 POLARIZATION CHANGES OWING TO STIMULATED RAMAN SCATTERING

We begin with a model consisting of a pulsar with a non-thermal component of radiation interacting with plasma in the emission region at a distance r 100RNS 108 cm (neutron star radius RNS < 10 km) from the neutron star surface, where the magnetic field is about 106 G. Plasma particles may all be in their lowest Landau level with no Larmor gyration; however, plasma can have one-dimensional distributions of momenta along the magnetic field (Blandford 1975; Lominadze, Machabeli & Usov 1983). This is because the synchrotron loss time for the decay of the perpendicular component of momentum is always short compared with the transit time at the stellar surface for any velocity. The simplest model for the plasma is density declining in proportion to r23 with no gradients in the distribution functions across B.

The non-linear interaction of radiation with plasma causes the modulation instability, leading to enhancement of non-resonant density perturbations and radiation amplification by free±electron±maser, which produces intense electromagnetic waves (Gangadhara & Krishan 1992; Gangadhara et al. 1993; Lesch, Gil & Shukla 1994). Since the frequency of these electromagnetic waves is close to the plasma frequency, they resonantly couple with the subpulse-associated plasma column in the pulsar magnetosphere and drive the stimulated Raman scattering.

Consider a large-amplitude electromagnetic wave (ki, v i) with an electric field

Ei 1xi cos ki´r 2 vite^x 1 1yi cos ki´r 2 vit 1 die^yY

1

which interacts with the plasma in the pulsar magnetosphere, where (eÃx, eÃy) are the unit vectors, (exi, eyi) are the x- and y- components of

electric field, and di is the initial phase.

We follow Ruderman & Sutherland's (1975) approach to estimate the density and plasma frequency of the plasma moving within the bundle of field lines. For typical parameters (Lorentz factor of primary particles gp , 106, and for pair plasma particles g^ , 103, magnetic field B0 , 106r28 3 G and pulsar period p 1 s), we obtain the particle number density

n0 gp B0 3X5 Â 107 cm23

2

2g^ ecp

r38

and the plasma frequency

4pn0e2 1a2 2 Â 1010

vp 2g^ g^m0

r3a2 rad s21Y

3

8

where e and m0 are the charge and rest mass of the electron, and c is the speed of light. The plasma in the emission region of pulsar magnetosphere may be birefringent (Melrose & Stoneham 1977; Melrose 1979; Barnard &

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832 R. T. Gangadhara, H. Lesch and V. Krishan

Arons 1986; von Hoensbroech, Lesch and Kunzl 1998). In these models, two modes of wave propagation are generally allowed in

magnetoactive plasma: one mode is polarized in the ki±B plane and other mode in the direction perpendicular to it. Following these models, we resolve Ei into two modes, Exi 1xi cos ki´r 2 vite^x and Eyi 1yi cos ki´r 2 vit 1 die^y, such that they are polarized in the directions parallel and perpendicular to the ki±B plane, respectively. It is well known from laser±plasma interactions that large-amplitude electromagnetic waves resonantly interact with the plasma when the radiation frequency becomes close to the plasma frequency. Since the

two modes have different indices of refraction (McKinnon 1997), i.e. the response of plasma is not the same for the two radiation modes, it is reasonable to assume that Exi couples with the density perturbation Dn1 dn1 cos k´r 2 vt, and Eyi couples with Dn2 dn2 cos k´r 2 vt 1 d in the plasma medium.

Since the ponderomotive force is proportional to 7E2i1 and 7E2i2, the coupling between the radiation and the density perturbations is non-linear. Hence the density perturbations grow, leading to currents and mixed electromagnetic±electrostatic sideband modes at ki ^ kY vi ^ v. In turn these sideband modes couple with the incident wave field, producing a much stronger ponderomotive force, which amplifies the original density perturbation. Hence a positive feedback system sets in, which leads to an instability.

The electric field Es of the electromagnetic wave scattered through an angle f s with respect to ki can be written as

Es 1xs cos ks´r 2 vste^Hx 1 1ys cos ks´r 2 vst 1 dse^HyX

4

The propagation directions of the incident wave (ki,v i) and the scattered wave (ks,v s) are illustrated in Fig. 1, such that ki k e^zY kskeÃ Hz and eÃ HykeÃy. The primed coordinate system is rotated through an angle f s about the y-axis. Then the scattered wave in the unprimed

coordinate system is given by

Es 1xs cos fs cos ks´r 2 vste^x 1 1ys cos ks´r 2 vst 1 dse^y 2 1zssin fs cos ks´r 2 vste^zY

5

where 1 zs=1 xs. The quiver velocity u^ of positrons and electrons owing to the radiation fields Ei and Es is given by

2u^ ^ e Ei 1 EsY

6

2t

m0

where e and m0 are the charge and rest mass of the plasma particle.

The wave equation for the scattered electromagnetic wave is given by

1 22 4p 2J

72 2 c2 2t2 Es c2 2t Y

7

where c is the velocity of light and J is the current density. The components of equation (7) are

Ds1xs cos fs cos ks´r 2 vst 2 2mp0e2 vvsi 1xidn1 cos ks´r 2 vstY 8

Ds1ys cos ks´r 2 vst 1 ds 2 2pe2 vs 1yidn2 cos ks´r 2 vst 1 di ÿ d

9

m0 vi

Figure 1. Stimulated Raman scattering of a transverse electromagnetic wave (Exi, Eyi) through an angle f s. The scattered wave electric field is (Exs, Eys). The wavenumbers ki, ks and k are due to the incident, scattered and Langmuir waves, respectively.

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Polarization of pulsar radio emission 833

and

Ds1zs sin fs cos ks´r 2 vst 0Y

10

where Ds k2s c2 2 v2s 1 v2p and

vs vi 2 vY ks ki 2 kX

11

In quantum language these two relations may be interpreted as the conservation of energy and the momentum along the magnetic field, respectively. When these conditions are satisfied, stimulated Raman scattering is excited resonantly and the expression Ds < 0 becomes the dispersion relation of the scattered electromagnetic mode.

If we cancel the instantaneous space- and time-dependent cosine functions on both sides of equation (8), we get

Ds1xs cos fs 2 2mp0e2 vvsi 1xidn1X 12

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Similar to the phase matching conditions (equation 11), equation (9) gives the condition between the initial phases:

ds di ÿ dY

13

and hence we obtain

Ds1ys 2 2pe2 vs 1yidn2X

14

m0 vi

Dividing equation (14) by equation (12), we have

as aihcos fsY

15

where ai 1yia1xi, as 1ysa1xs and h dn2adn1. The value of h is determined by Ei in such a way that Dn1 couples with Exi and Dn2 couples with Eyi.

We consider the Vlasov equation to find the low-frequency plasma response:

2f 1 v´7f 1 1 e7f 2 7c´ 2f 0Y

16

2t

m0

2v

where f(r,t) is the scalar potential associated with the electrostatic waves, f(r,v,t) is the particle distribution function and c (r,t) is the

ponderomotive potential.

Using f rY vY t f 0 v 1 Df 1 rY vY t 1 Df 2 rY vY t, we can linearize equation (16), and obtain

2 Df 1 1 2 Df 2 1 v´7 Df 1 v´7 Df 1 1 e7f 2 7c´ 2f 0 0Y

17

2t

2t

1

2 m0

2v

where Df 1 df 1cos k´r 2 vt and Df 2 df 2cos k´r 2 vt 1 d. The ponderomotive force of the radiation field depends quadratically on the amplitude of the electric field. Physically, it is a radiation pressure which amplifies the density perturbations by exciting the slow

longitudinal fields and motions. The ponderomotive potential is given by

e2 B E E 2C

e2

c 2m

Reivi

1

s

iv

2m v2 cos fs cos k´r 2 vt 1 aias cos k´r 2 vt 1 di 2 ds1i1sY

18

0

i

s

v

0i

where the brackets k lv represent the average over the fast time-scale (vi @ v). In the presence of a strong magnetic field, the pair plasma becomes polarized and one-dimensional. If there is some relativistic drift

between the electrons and the positrons then the ponderomotive force becomes effective, which will induce the non-linear density perturbations (AsseÂo 1993):

1 j1kj2 v2p 1

dn^ < 2 32p kBTp v2i g^ Y

19

where 1 k is the envelope of the parallel electric field of the Langmuir wave which is slowly varying with space and time, and Tp is the plasma temperature. Using these density perturbations and Poisson equation, we can self-consistently derive the potential f :

f 2 4kp2e Dn1 1 Dn2X 20

Now, substituting the expressions for Df1, Df2, Dn1, Dn2, f and c into equation (17) and using the condition (13), we obtain

&4pe2

e2

' k´ 2f 0 2v

df 2 1 df 1m 1 m0k2 dn2 1 dn1m 1 2m20v2i cos fsm 1 aias1i1s v 2 k´v 0Y

21

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834 R. T. Gangadhara, H. Lesch and V. Krishan

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where m sin k´r 2 vtasin k´r 2 vt 1 d. For d 0 and pY m ^1, while at the other values of d we have to find the average kml ma over the time-scale T 2pav. Therefore we have

4pe2

1ik2

k´ 2f 0 2v

df 2 1 df 1ma 2 m0k2

dn2 1 dn1ma 1 8pm0v2 A v 2 k´v Y

i

22

where A macosfs 1 aias1s. The sum of density perturbations (dn2 1 dn1ma) can be estimated as

1ik2 !

dn2 1 dn1ma 121n0 df 2 1 df 1madv ÿ dn2 1 dn1ma 1 8pm0v2 A xY

23

i

where

v2 1 k´ 2f 0

x p

2v dv

24

k2 21 v ÿ k´v

is the susceptibility function (Fried & Conte 1961; Liu & Kaw 1976). Using h dn2adn1, we can write

1

1ik2

1 1 x h 1 madn1 2 8pm0v2 AX

25

i

Multiplying equation (12) by 1 im a and equation (14) by a i1 i, and adding, we get

asai 1 m cosf 1s 2 2pe2 vs 1i m 1 a2h dn1 Y

26

a

s

m0 vi

a

i Ds

where 1i 1xi and 1s 1xs.

Now, using equations (25) and (26), we obtain the dispersion relation for stimulated Raman scattering:

1

v20k2 ma 1 a2i h vs 1

1 1 x h 1 ma 4 1 1 a2 vi Ds Y

27

i

p

where v0 e1i 1 1 a2i am0vi is the quiver velocity of plasma particles owing to the electric field of the incident electromagnetic wave.

If L L30 Â 1030ergs21 is the luminosity of pump radiation with frequency vi 2pni n10 Â 1010 rad s21 at a distance r

r8 Â 108 cm from the source then

e 2L 1a2

L1a2

v0 m0vi r2c

4X3 Â 109 30 cm s21X n10r8

28

When the phase matching conditions are met, the instability becomes more efficient, and Ds 2viv 2 v2 2 2c2ki´k 1 c2k2 < 0 represents

the dispersion relation of the Stokes mode. The thermal speed vt of the plasma can be expressed in terms of its energy spread in the laboratory frame. Let v^ c 1 2 1ag2^1a2 be

the velocity of the electron±positron plasma; then the velocity spread d v^ is given by (Hasegawa 1978; Gangadhara & Krishan 1992)

dv^ < c dgg3^ for g @ 1X 29

Now, using the Lorentz transformation of velocity, we can show that

dv^ dvz

dvz H

< dvz H vt

30

g2^ 1 1 v^vz Hac2 g2 g2

because vz H 0. Hence the thermal speed, in the plasma frame, is given by

vt c dg^ X

31

g^

For g^ 103, we get vt 3 Â 107dg^ cm s21. If we separate equation (27) into real and imaginary parts, we get two coupled simultaneous equations. By solving them numerically,

we find the growth rate G of stimulated Raman scattering. Fig. 2 shows the growth rate as a function of r8 and vi/vp. Since the plasma density decreases with r as 1/r3, the growth rate decreases as r8 increases. Also, if the radiation frequency becomes higher than the plasma frequency, then radiation and plasma do not couple resonantly, which leads to the decrease of the growth rate at higher values of vi/vp.

To study the variation of G in the a i±d g^ plane, we have made a contour plot (Fig. 3); the labels on the contours indicate the values of log (G). The Debye length increases with the increase of d g^, therefore the plasma wave experiences a large Landau damping, which leads to the drop in growth rate. The parameter a i is the ratio of amplitudes of electromagnetic waves that are polarized in the directions parallel and perpendicular to the ki±B plane. When a i is small, the wave polarized in the direction parallel to the ki±B plane becomes strong and

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Polarization of pulsar radio emission 835

will have a component along B. Therefore the coupling between radiation and plasma will be strong, which leads to the higher growth rate

at the smaller values of ai. For v ! c2ki´kavi, Stokes mode becomes resonant, and the anti-Stokes mode becomes non-resonant. Then equation (27) can be

written as

v 2 vl 1 iGl v 2 vl 1 iGs h 1 m 21018 L30 ma 1 a2i h rad2 s22Y

32

a

r78a2n10 1 1 a2i

where v2l v2p 1 3a2k2v2t ,

p

!

Gl p vp exp 2 1 2 3 1 nc

33

2 klD3

2 klD2 2

is the damping rate of the plasma wave, and the Debye length lD vtap2vp. For klD , 1, we find Gl < 2X5 Â 109ar38a2s21. The pair plasma collision frequency nc < 2X5 Â 1023 ln Lar38dg3^ s21 and the Coulomb logarithm ln L < 10. The collisional damping rate of the scattered electromagnetic wave is given by Gs v2pnca2v2s < 0X06an210r68dg3^s21.

For klD . 1, plasma waves are highly damped and hence the plasma loses its collective behaviour. Therefore stimulated Raman

scattering changes into induced Compton scattering of electromagnetic waves by the plasma particles. In the conventional treatment of

induced Compton scattering in pulsars (e.g. Blandford & Scharlemann 1976; Sincell & Krolik 1992) the collective effects of the plasma are

ignored. The collective treatment of the wave scattering by plasma particles is justified if the condition klD ! 1 or vph @ vt is met, where

vph wpak is the phase velocity of the plasma wave. This condition implies that the wavenumber of oscillation of the electrons in the beat

wave of the incident and scattered waves is much less than the inverse of the Debye length. It is not satisfied when the beat wave is strongly

Landau damped (klD . 1) or the beat wave will not feel the presence of a medium ([email protected]), so that the scattering process will be described

as induced Compton scattering.

When klD ! 1, by setting v vl 1 iG, we can solve equation (32) for the growth rate:

s

G 2 1 Gl 1 Gs ^ 1 Gl ÿ Gs2 1 4X3 Â 1018 L30 ma 1 a2i h X

34

2

2

r78a2n10 1 1 a2i h 1 ma

Stimulated Raman scattering is a threshold process: if the intensity of the pump exceeds the threshold, only then will it start converting its

energy into decay waves. The threshold condition for the excitation of stimulated Raman scattering is given by

L30

219 3a2 1 1 a2i h 1 ma

r2 9X3 Â 10 n10r8 GlGs m 1 a2h X

35

8 thr

a

i

The typical threshold intensities for stimulated Raman scattering are of the order of the observed intensities, implying that pulsar magnetosphere may be optically thick to Raman scattering of electromagnetic waves.

Figure 2. The growth rate G of stimulated Raman scattering is plotted with respect to r8 and v i/vp.

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Figure 3. A contour plot of G in the d^±a i plane. The labels on the contours indicate the log (G) values.

836 R. T. Gangadhara, H. Lesch and V. Krishan

The growth rate just above the threshold is given by

G 4X3 Â 108 L30 ma 1 a2i h s21Y

36

r28n10 1 1 a2i h 1 ma

which is proportional to L30. The maximum growth rate attainable for vp . G . Gl, on the other hand, is

s

G 109 L30

ma 1 a2i h s21X

37

n310r183a2 1 1 a2i h 1 ma

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3 STOKES PARAMETERS

When the phase matching conditions are satisfied, the growth rate G becomes large, and the scattered mode is amplified and becomes a normal mode of the plasma. Under some conditions, the scattered mode leaves the plasma with a polarization which may be different from the polarization of the incident wave. The polarization states of the incident and scattered waves can be described more accurately using the Stokes parameters (Rybicki & Lightman 1979):

Ij 1xj1Ãxj 1 1yj1ÃyjY

38

Qj 1xj1Ãxj 2 1yj1ÃyjY

39

Uj 21xj1Ãyj cos dj

40

and

Vj 21xj1Ãyj sin djY

41

where j i for the incident wave, and s for the scattered wave. The linear polarization is given by

q

L U2j 1 Q2j Y

42

and the polarization position angle is given by

xj 12 arctan UjaQjX 43

The transfer of energy between the modes will be efficient only when the energy of the pump wave is strong enough to overcome the damping losses or escape of the generated waves. Using the Manley±Rowe relation (Weiland & Wilhelmsson 1977)

Ii Is Y

44

vi vs

we find the relation between the incident flux Ii and the scattered flux Is:

v

Is 1 2 vi IiX

45

In the following two subsections, we consider the cases in which the polarization state of the incident wave is linear and circular, and compute the polarization states of the scattered mode.

3.1 Linearly polarized incident wave

Consider a linearly polarized electromagnetic wave (di 0, ai 10 and xi 908), which excites stimulated Raman scattering in the magnetospheric plasma. Using the plasma and magnetic field parameters discussed in the previous section, we compute the growth rate of stimulated Raman scattering. For klD ! 1, the instability becomes quite strong and the Landau damping of the Langmuir wave is minimal. Hence the stimulated Raman scattering is resonantly excited. Fig. 4 shows the behaviour of linear (solid line) and circular (broken line) polarization of the scattered mode with respect to h . It shows that, at some values of h which are close to 0.1, the linear polarization of the incident wave can be converted almost completely into circular polarization of the scattered wave. Charge density variations within the pulse window could cause the conversion efficiency of linear to circular polarization to vary.

The variation of the polarization angle of the scattered wave with respect to h , at different values of d , is shown in Fig. 5. For d 908 and h # 0X1, the scattered mode becomes orthogonally polarized with respect to the incident wave. Furthermore, if there is any variation in the plasma density or radiation intensity, the value of h fluctuates and the scattered modes produced with h # 0X1 become orthogonally polarized with respect to those produced with h . 0X1.

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Polarization of pulsar radio emission 837

3.2 Circularly polarized incident wave

Suppose that the incident wave is circularly polarized (di pa2 and ai 1); then the scattered mode will be linearly polarized for h # 0X2, while, at the other values, both linear and circular polarizations with different proportions will exist, as indicated by Fig. 6. For di , 0 and ds di 2 d , 0 the sense of circular polarization of the scattered mode becomes opposite to that of the incident wave. Hence, depending upon the plasma and radiation conditions, stimulated Raman scattering can change the polarization of the pulsar radio signals.

4 DISCUSSION

The variable nature of circular polarization is evident from the polarization distributions of Manchester, Taylor & Huguenin (1975), Backer & Rankin (1980)and Stinebring et al. (1984a,b). Very high degrees of circular polarization are occasionally observed in individual pulses, even up to 100 per cent (Cognard et al. 1996), while the integrated or average pulse profiles generally indicate a much smaller degree of circular polarization (e.g. Lyne, Smith & Graham 1971), which show that the sign of circular polarization is variable at any given pulse phase. Radhakrishnan & Rankin (1990) have identified two extreme types of circular polarization in the observations: (i) an antisymmetric type wherein the circular polarization changes sense in mid-pulse; and (ii) a symmetric type wherein it is predominantly of one sense. The correlation of sense of the antisymmetric type of circular polarization with the polarization angle swing indicates the geometric property of the emission process, and is highly suggestive of curvature radiation.

The diverse nature of circular polarization may be the consequence of the pulsar emission mechanism and the subsequent propagation effects in the pulsar magnetosphere (e.g. Melrose 1995; von Hoensbroech, Lesch & Kunzl 1998). It seems rather difficult to explain the various circular polarization behaviours using the widely accepted magnetic pole models (e.g. Radhakrishnan & Cooke 1969; Komesaroff 1970; Sturrock 1971; Ruderman & Sutherland 1975).

The symmetric type of circular polarization observed in some pulsars (e.g. PSR B1914113 and B0628±28), as shown in Figs 7 and 8, may be due to the propagation effects. However, it seems difficult for propagation effects to explain how the sign of the circular polarization can change precisely at the centre of the pulse in the case of the antisymmetric type, as seen in many pulsars, e.g. PSR B1859103 and B1933116 (Rankin, Stinebring & Weisberg 1989).

If we are to understand the radio emission mechanism, we must understand the physical state of the radio-loud plasma in the polar cap. It is this plasma that is the site of instabilities which are thought to produce coherent radio emission. The role of different propagation effects in the pulsar polarization has been discussed by Cheng & Ruderman (1979), Beskin, Gurevich & Istomin (1988) and Istomin (1992). The mechanisms proposed by these authors predict a frequency dependence for circular polarization, with weaker polarization at higher frequencies. This is seen in some pulsars (e.g. PSR B0835±41, B1749±28 and B1240±64), but it is not generally the case (Han et al. 1998). Istomin suggested that the linearly polarized incident wave becomes circularly polarized as a result of generalized Faraday rotation; however, it is observationally known that no generalized Faraday rotation is evident in pulsar magnetospheres (Cordes 1983; Lyne & Smith 1990).

We have presented a model to explain the polarization changes owing to the propagation of radio waves through the magnetospheric

Figure 4. The solid line and broken curves indicate the variation of linear (Ls) and circular (Vs) polarization of the scattered mode with respect to h. The normalizing parameter Is is the intensity of the scattered mode.

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Figure 5. The polarization angle xs of the scattered mode is plotted as a function of h, at different values of d (08, 308, 608 and 908).

838 R. T. Gangadhara, H. Lesch and V. Krishan

Figure 6. The variation of linear (Ls) and circular (Vs) polarization of the scattered mode, with respect to h, indicated by the solid line and broken curves, respectively. The normalizing parameter Is is the intensity of the scattered mode.

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Figure 7. PSR B0628±28, an example of a pulsar with `symmetric' circular polarization and high linear polarization (from Lyne & Manchester 1988).

Figure 8. The polarization of PSR B1914+13, a pulsar with strong circular polarization over the whole observed profile (from Rankin et al. 1989).

plasma. The features, like a large change in polarization angle, a sense reversal of circular polarization and extremely rapid temporal changes in intensity, would help us to explain many observations, for which the existing mechanisms have proved to be inadequate. Because of the very strong dependence of polarization angle on plasma parameters via the growth rate, in an inhomogeneous plasma medium the depolarization is a natural outcome. We believe that plasma processes such as the stimulated Raman scattering may be a potential mechanism for the polarization variability in pulsars. The circular polarization of a number of pulsars varies with frequency. The two clear

q 1999 RAS, MNRAS 307, 830±840

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Polarization of pulsar radio emission 839

examples are PSR B1240±64 and B2048±78, from which the opposite senses or transitions of circular polarization have been observed at lower and higher frequencies in the radio band (Han et al. 1998).

5 CONCLUSION

We have considered the stimulated Raman scattering of the transverse electromagnetic waves in the electron±positron plasma of a pulsar magnetosphere. Since the frequency of the incident wave is much higher than the plasma frequency in stimulated Raman scattering, the change in scattered wave frequency is very small compared with the incident wave frequency. The value of the radiation±plasma coupling parameter h is determined by the polarization of the incident wave, and its value can be determined only by non-linear analysis. In the process of three-wave interaction, the phase matching condition (equation 13) between the initial phases (d i, d s, d ) and the value of h determine the polarization state of the scattered wave.

Many short-time-scale polarization variabilities in individual pulses, such as variations in the amount of linear and circular polarization, a sense reversal of circular polarization, and polarization angle swings, can be accounted for by considering stimulated Raman scattering. The time-scales over which the changes take place in individual pulses are of the order of the e-folding time, which is the inverse of the growth rate of stimulated Raman scattering.

It seems rather unlikely that the diverse behaviour of circular polarization can be accounted for by a single mechanism. Both intrinsic emission and propagation effects seem possible. The strong symmetric type of circular polarization observed in some pulsars is most probably generated by propagation effects, such as stimulated Raman scattering. Further simultaneous observations over a wide frequency range would be valuable in sorting out the importance of propagation effects.

ACKNOWLEDGMENTS

We are grateful to Y. Gupta and A. von Hoensbroech for discussions and comments. Fig. 8 has been reproduced, with permission, from Astrophysical Journal, published by The University of Chicago Press. (q 1989 by the American Astronomical Society. All rights reserved.) We thank Dr. J. M. Rankin for allowing us to use Fig. 8.

REFERENCES

AsseÂo E., 1993, MNRAS, 264, 940 AsseÂo E., Pelletier G., Sol H., 1990, MNRAS, 247, 529 Backer D. C., Rankin J. M., 1980, ApJS, 42, 143 Barnard J. J., Arons J., 1986, ApJ, 302, 138 Beal J. H., 1990, in Brinkmann W., Fabian A. C., Giovannelli F., eds, Physical Processes in Hot Cosmic Plasmas. Kluwer, Dordrecht, p. 341 Benford G., 1992, ApJ, 391, L59 Beskin V. S., Gurevich A. V., Istomin Ya. N., 1988, Ap. C&SS, 146, 205 Blandford R. D., Scharlemann E. T., 1976, MNRAS, 174, 59 Blandford R. D., 1975, MNRAS, 170, 551 Cheng A. F., Ruderman M. A., 1979, ApJ, 229, 348 Cognard I., Shrauner J. A., Taylor J. H., Thorsett S. E., 1996, ApJ, 457, L81 Cordes J. M., 1979, Space Sci. Rev., 24, 567 Cordes J. M., 1983, in Burns M. L., Harding A. K., Ramaty R., eds, AIP Conf. Proc. 101, Positron±Electron pairs in Astrophysics. Am. Inst. Phys., New

York, p. 98 Drake J. F., Kaw P. K., Lee Y. C., Schmidt G., Liu C. S., Rosenbluth M. N., 1974, Phys. Fluids, 17, 778 Fried D., Conte S. D., 1961, The Plasma Dispersion Function. Academic Press, New York, p. 1 Gangadhara R. T., Krishan V., 1992, MNRAS, 256, 111 Gangadhara R. T., Krishan V., 1993, ApJ, 415, 505 Gangadhara R. T., Krishan V., 1995, ApJ, 440, 116 Gangadhara R. T., Krishan V., Shukla P. K., 1993, MNRAS, 262, 151 Han J. L., Manchester R. N., Xu R. X., Qiao G. J., 1998, MNRAS, 300, 373 Hasegawa A., 1978, Bell System Tech. J., 57, 3069 Istomin Ya. N., 1992, in Hankins T. H., Rankin J. M., Gil J. A., eds, Proc. IAU Colloq. 128, The magnetosphere structure and emission mechanism of radio

pulsars. Pedagogical Univ. Press, Zielona GoÂra, p. 375 Komesaroff M. M., 1970, Nat, 225, 612 Krishan V., Wiita P. J., 1990, MNRAS, 246, 597 Kruer W. L., 1988, The Physics of Laser±plasma interactions. Addison-Wesley, New York, p. 70 Lesch H., Gil J. A., Shukla P. K., 1994, Space Sci. Rev., 68, 349 Liu C.S., Kaw P.K. 1976, in Simon, A., Thompson W. B., eds, Advances in Plasma Physics. Interscience, New York, Vol. 6, p. 83 Lominadze Dzh. G., Machabeli G. Z., Usov V. V. 1983, ApS&S, 90, 19 Lyne A. G., Manchester R. N., 1988, MNRAS, 234, 477 Lyne A. G., Smith F. G., 1990, Pulsar Astronomy, Cambridge Univ. Press, Cambridge, 234 Lyne A. G., Smith F. G., Graham D. A., 1971, MNRAS, 153, 337 McKinnon 1997, ApJ, 475, 763 Manchester R. N., Taylor J. H., Huguenin G. R., 1975, ApJ, 196, 83

q 1999 RAS, MNRAS 307, 830±840

Mon. Not. R. Astron. Soc. 307, 830±840 (1999)

Propagation effects on polarization of pulsar radio emission

R. T. Gangadhara,1,2 H. Lesch3 and V. Krishan1

1Indian Institute of Astrophysics, Bangalore 560034, India 2National Centre for Radio Astrophysics, TIFR, Pune University Campus, Pune 411007, India 3Institut fuÈr Astronomie und Astrophysik der UniversitaÈt MuÈnchen, Scheinerstrasse 1, 81679 MuÈnchen, Germany

Accepted 1999 March 19. Received 1999 March 19; in original form 1998 May 6

ABSTRACT We consider the role of a propagation effect such as stimulated Raman scattering on the polarization of radio pulses. When an intense electromagnetic wave with frequency close to the plasma frequency interacts with the plasma in the pulsar magnetosphere, the incident wave undergoes stimulated Raman scattering. Using typical plasma and magnetic field parameters, we compute the growth rate and estimate the polarization properties of the scattered mode. Under some conditions, we find that the polarization properties of the scattered mode can become significantly different from those of the incident wave. The frequencies at which strong Raman scattering occurs in the outer parts of the magnetosphere fall into the observable radio band. In some pulsars, for example PSR B0628228 and B1914113, a large and symmetric type of circular polarization has been observed. We propose that such an unusual circular polarization is produced by the propagation effects.

Key words: plasmas ± polarization ± radiative transfer ± waves ± stars: magnetic fields ± pulsars: general.

1 INTRODUCTION

The investigation of coherent radio emission from pulsar magnetospheres has attracted a great deal of attention (Cordes 1979; Michel 1982; AsseÂo, Pelletier & Sol 1990). Powerful collective emission occurs when relativistic electron beams with density ,1 per cent of the pair plasma density scatter coherently from concentrations of plasma waves (cavitons) (Benford 1992). The role of collective plasma processes in the absorption and spectral modification of radio waves is well known (e.g. Beal 1990; Krishan & Wiita 1990; Benford 1992; Gangadhara & Krishan 1992, 1993, 1995; Gangadhara, Krishan & Shukla 1993). von Hoensbroech, Lesch and Kunzl (1998) have demonstrated that degree of linear polarization decreases with increasing frequency while the degree of circular polarization shows the opposite trend.

In this paper, we estimate the role of stimulated Raman scattering in the polarization of electromagnetic waves propagating in the pulsar magnetosphere. We assume that the physical conditions in the pulsar magnetosphere are those of the classical standard model (Ruderman & Sutherland 1975) which describes the generation of ultrarelativistic beams of electrons/positrons and the creation of the pair plasma. The beams and the pair plasma are in relativistic motion along the bundle of open magnetic field lines that delimit the active region of the magnetosphere. The stimulated Raman scattering is considered as a parametric decay of the initial transverse electromagnetic (pump) wave into another electromagnetic wave and a longitudinal plasma wave. The physics of stimulated Raman scattering in a plasma has been explained in many papers and books (e.g. Drake et al. 1974; Liu & Kaw 1976; Hasegawa 1978; Kruer 1988). There are two ways in which stimulated Raman scattering may be important in the pulsar environment. First, it may act as an effective damping mechanism for the electromagnetic waves generated by some emission mechanism at the lower altitudes in the pulsar magnetosphere. At those altitudes, the resonant conditions for stimulated Raman scattering, i.e. frequency and wavenumber matching, might not be satisfied. This results in a short time-scale of variability which is generally observed in pulsar radio emission. Secondly, stimulated Raman scattering may provide an effective saturation mechanism for the growth of the electromagnetic waves, provided that the conditions for wave excitation by some mechanism are satisfied in the region where effective stimulated Raman scattering can take place.

The first case can be simplified by treating the intensity of the pump as constant in time. Then the non-linear equations, which describe the wave coupling, become linear in the amplitudes of the decaying waves, and the exponentially growing solutions imply an effective energy transfer from the pump wave. This approximation breaks down when the amplitudes of the decay waves become comparable to that of the pump wave or when the amplitudes of the decay waves enter the non-linear stage, and start losing energy owing to some non-linear processes such as wave±particle trapping and acceleration.

The second case is more complicated, where stimulated Raman scattering acts as a non-linear saturation mechanism and the

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Polarization of pulsar radio emission 831

amplitudes of all waves may be of the same order. This case can be considerably simplified when damping of the plasma wave is very strong or if it leaves the region of the resonant interaction quickly enough.

We neglect the non-linear stages of stimulated Raman scattering and the development of Langmuir turbulence, which leads to waveparticle trapping or quasi-linear diffusion. If the pump is monochromatic, the growth rate of stimulated Raman scattering can become very high, as in conventional laboratory laser±plasma interaction. However, in the case of pulsars the pump can be broadband, and, in the limit where the bandwidth Dv of the pump wave is much larger than the growth rate of stimulated Raman scattering, we can use a random phase approximation for the statistical description of the interacting waves.

Tsytovich & Shvartsburg (1966) have given a general expression for the third-order non-linear currents excited in a magnetized plasma. Since the corresponding expressions are very complicated, the general case of Raman scattering becomes very difficult to consider. However, one can make some useful simplifications when considering stimulated Raman scattering in the pulsar magnetosphere. First, in a superstrong magnetic field, we can expand the currents in terms of 1/v B, where vB is the cyclotron frequency. Secondly, if the pair plasma has the same distributions for electrons and positrons then some of the non-linear currents cancel out, as they are proportional to the third power of the electric charge (this cancellation is exact in the unmagnetized electron±positron plasma). Thirdly, all the three interacting waves propagate along the magnetic field. This is an important but less justified approximation. It allows us to simplify the problem considerably, and to obtain a dispersion relation for stimulated Raman scattering.

The polarization of the pulsar signals appear to bear critically on the pulsar radio emission process and the emission beam geometry. One or more reversals of the sense of circular polarization have been observed in the intergrated profiles of several pulsars. However, in individual pulses, circular polarization changes sense many times across the pulse window. Further, it is important to determine whether the depolarization is a geometric effect or results from radiation±plasma interactions. There have only been very preliminary attempts to explain depolarization and microvariability using plasma mechanisms (Benford 1992).

Our purpose in this paper is to derive the properties of natural plasma modes and to explore some possible implications concerning the interpretation of the observed polarization, notably large and symmetric circular polarization in some pulsars (e.g. PSR B0628±28 and B1914113). In Section 2, we derive the dispersion relation for stimulated Raman scattering of an electromagnetic wave, and give an analytical expression for the growth rate of the instability. In Section 3, we define the Stokes parameters and compute the polarization states of the scattered electromagnetic wave. The discussion of our findings is given in Section 4.

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2 POLARIZATION CHANGES OWING TO STIMULATED RAMAN SCATTERING

We begin with a model consisting of a pulsar with a non-thermal component of radiation interacting with plasma in the emission region at a distance r 100RNS 108 cm (neutron star radius RNS < 10 km) from the neutron star surface, where the magnetic field is about 106 G. Plasma particles may all be in their lowest Landau level with no Larmor gyration; however, plasma can have one-dimensional distributions of momenta along the magnetic field (Blandford 1975; Lominadze, Machabeli & Usov 1983). This is because the synchrotron loss time for the decay of the perpendicular component of momentum is always short compared with the transit time at the stellar surface for any velocity. The simplest model for the plasma is density declining in proportion to r23 with no gradients in the distribution functions across B.

The non-linear interaction of radiation with plasma causes the modulation instability, leading to enhancement of non-resonant density perturbations and radiation amplification by free±electron±maser, which produces intense electromagnetic waves (Gangadhara & Krishan 1992; Gangadhara et al. 1993; Lesch, Gil & Shukla 1994). Since the frequency of these electromagnetic waves is close to the plasma frequency, they resonantly couple with the subpulse-associated plasma column in the pulsar magnetosphere and drive the stimulated Raman scattering.

Consider a large-amplitude electromagnetic wave (ki, v i) with an electric field

Ei 1xi cos ki´r 2 vite^x 1 1yi cos ki´r 2 vit 1 die^yY

1

which interacts with the plasma in the pulsar magnetosphere, where (eÃx, eÃy) are the unit vectors, (exi, eyi) are the x- and y- components of

electric field, and di is the initial phase.

We follow Ruderman & Sutherland's (1975) approach to estimate the density and plasma frequency of the plasma moving within the bundle of field lines. For typical parameters (Lorentz factor of primary particles gp , 106, and for pair plasma particles g^ , 103, magnetic field B0 , 106r28 3 G and pulsar period p 1 s), we obtain the particle number density

n0 gp B0 3X5 Â 107 cm23

2

2g^ ecp

r38

and the plasma frequency

4pn0e2 1a2 2 Â 1010

vp 2g^ g^m0

r3a2 rad s21Y

3

8

where e and m0 are the charge and rest mass of the electron, and c is the speed of light. The plasma in the emission region of pulsar magnetosphere may be birefringent (Melrose & Stoneham 1977; Melrose 1979; Barnard &

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832 R. T. Gangadhara, H. Lesch and V. Krishan

Arons 1986; von Hoensbroech, Lesch and Kunzl 1998). In these models, two modes of wave propagation are generally allowed in

magnetoactive plasma: one mode is polarized in the ki±B plane and other mode in the direction perpendicular to it. Following these models, we resolve Ei into two modes, Exi 1xi cos ki´r 2 vite^x and Eyi 1yi cos ki´r 2 vit 1 die^y, such that they are polarized in the directions parallel and perpendicular to the ki±B plane, respectively. It is well known from laser±plasma interactions that large-amplitude electromagnetic waves resonantly interact with the plasma when the radiation frequency becomes close to the plasma frequency. Since the

two modes have different indices of refraction (McKinnon 1997), i.e. the response of plasma is not the same for the two radiation modes, it is reasonable to assume that Exi couples with the density perturbation Dn1 dn1 cos k´r 2 vt, and Eyi couples with Dn2 dn2 cos k´r 2 vt 1 d in the plasma medium.

Since the ponderomotive force is proportional to 7E2i1 and 7E2i2, the coupling between the radiation and the density perturbations is non-linear. Hence the density perturbations grow, leading to currents and mixed electromagnetic±electrostatic sideband modes at ki ^ kY vi ^ v. In turn these sideband modes couple with the incident wave field, producing a much stronger ponderomotive force, which amplifies the original density perturbation. Hence a positive feedback system sets in, which leads to an instability.

The electric field Es of the electromagnetic wave scattered through an angle f s with respect to ki can be written as

Es 1xs cos ks´r 2 vste^Hx 1 1ys cos ks´r 2 vst 1 dse^HyX

4

The propagation directions of the incident wave (ki,v i) and the scattered wave (ks,v s) are illustrated in Fig. 1, such that ki k e^zY kskeÃ Hz and eÃ HykeÃy. The primed coordinate system is rotated through an angle f s about the y-axis. Then the scattered wave in the unprimed

coordinate system is given by

Es 1xs cos fs cos ks´r 2 vste^x 1 1ys cos ks´r 2 vst 1 dse^y 2 1zssin fs cos ks´r 2 vste^zY

5

where 1 zs=1 xs. The quiver velocity u^ of positrons and electrons owing to the radiation fields Ei and Es is given by

2u^ ^ e Ei 1 EsY

6

2t

m0

where e and m0 are the charge and rest mass of the plasma particle.

The wave equation for the scattered electromagnetic wave is given by

1 22 4p 2J

72 2 c2 2t2 Es c2 2t Y

7

where c is the velocity of light and J is the current density. The components of equation (7) are

Ds1xs cos fs cos ks´r 2 vst 2 2mp0e2 vvsi 1xidn1 cos ks´r 2 vstY 8

Ds1ys cos ks´r 2 vst 1 ds 2 2pe2 vs 1yidn2 cos ks´r 2 vst 1 di ÿ d

9

m0 vi

Figure 1. Stimulated Raman scattering of a transverse electromagnetic wave (Exi, Eyi) through an angle f s. The scattered wave electric field is (Exs, Eys). The wavenumbers ki, ks and k are due to the incident, scattered and Langmuir waves, respectively.

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Polarization of pulsar radio emission 833

and

Ds1zs sin fs cos ks´r 2 vst 0Y

10

where Ds k2s c2 2 v2s 1 v2p and

vs vi 2 vY ks ki 2 kX

11

In quantum language these two relations may be interpreted as the conservation of energy and the momentum along the magnetic field, respectively. When these conditions are satisfied, stimulated Raman scattering is excited resonantly and the expression Ds < 0 becomes the dispersion relation of the scattered electromagnetic mode.

If we cancel the instantaneous space- and time-dependent cosine functions on both sides of equation (8), we get

Ds1xs cos fs 2 2mp0e2 vvsi 1xidn1X 12

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Similar to the phase matching conditions (equation 11), equation (9) gives the condition between the initial phases:

ds di ÿ dY

13

and hence we obtain

Ds1ys 2 2pe2 vs 1yidn2X

14

m0 vi

Dividing equation (14) by equation (12), we have

as aihcos fsY

15

where ai 1yia1xi, as 1ysa1xs and h dn2adn1. The value of h is determined by Ei in such a way that Dn1 couples with Exi and Dn2 couples with Eyi.

We consider the Vlasov equation to find the low-frequency plasma response:

2f 1 v´7f 1 1 e7f 2 7c´ 2f 0Y

16

2t

m0

2v

where f(r,t) is the scalar potential associated with the electrostatic waves, f(r,v,t) is the particle distribution function and c (r,t) is the

ponderomotive potential.

Using f rY vY t f 0 v 1 Df 1 rY vY t 1 Df 2 rY vY t, we can linearize equation (16), and obtain

2 Df 1 1 2 Df 2 1 v´7 Df 1 v´7 Df 1 1 e7f 2 7c´ 2f 0 0Y

17

2t

2t

1

2 m0

2v

where Df 1 df 1cos k´r 2 vt and Df 2 df 2cos k´r 2 vt 1 d. The ponderomotive force of the radiation field depends quadratically on the amplitude of the electric field. Physically, it is a radiation pressure which amplifies the density perturbations by exciting the slow

longitudinal fields and motions. The ponderomotive potential is given by

e2 B E E 2C

e2

c 2m

Reivi

1

s

iv

2m v2 cos fs cos k´r 2 vt 1 aias cos k´r 2 vt 1 di 2 ds1i1sY

18

0

i

s

v

0i

where the brackets k lv represent the average over the fast time-scale (vi @ v). In the presence of a strong magnetic field, the pair plasma becomes polarized and one-dimensional. If there is some relativistic drift

between the electrons and the positrons then the ponderomotive force becomes effective, which will induce the non-linear density perturbations (AsseÂo 1993):

1 j1kj2 v2p 1

dn^ < 2 32p kBTp v2i g^ Y

19

where 1 k is the envelope of the parallel electric field of the Langmuir wave which is slowly varying with space and time, and Tp is the plasma temperature. Using these density perturbations and Poisson equation, we can self-consistently derive the potential f :

f 2 4kp2e Dn1 1 Dn2X 20

Now, substituting the expressions for Df1, Df2, Dn1, Dn2, f and c into equation (17) and using the condition (13), we obtain

&4pe2

e2

' k´ 2f 0 2v

df 2 1 df 1m 1 m0k2 dn2 1 dn1m 1 2m20v2i cos fsm 1 aias1i1s v 2 k´v 0Y

21

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834 R. T. Gangadhara, H. Lesch and V. Krishan

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where m sin k´r 2 vtasin k´r 2 vt 1 d. For d 0 and pY m ^1, while at the other values of d we have to find the average kml ma over the time-scale T 2pav. Therefore we have

4pe2

1ik2

k´ 2f 0 2v

df 2 1 df 1ma 2 m0k2

dn2 1 dn1ma 1 8pm0v2 A v 2 k´v Y

i

22

where A macosfs 1 aias1s. The sum of density perturbations (dn2 1 dn1ma) can be estimated as

1ik2 !

dn2 1 dn1ma 121n0 df 2 1 df 1madv ÿ dn2 1 dn1ma 1 8pm0v2 A xY

23

i

where

v2 1 k´ 2f 0

x p

2v dv

24

k2 21 v ÿ k´v

is the susceptibility function (Fried & Conte 1961; Liu & Kaw 1976). Using h dn2adn1, we can write

1

1ik2

1 1 x h 1 madn1 2 8pm0v2 AX

25

i

Multiplying equation (12) by 1 im a and equation (14) by a i1 i, and adding, we get

asai 1 m cosf 1s 2 2pe2 vs 1i m 1 a2h dn1 Y

26

a

s

m0 vi

a

i Ds

where 1i 1xi and 1s 1xs.

Now, using equations (25) and (26), we obtain the dispersion relation for stimulated Raman scattering:

1

v20k2 ma 1 a2i h vs 1

1 1 x h 1 ma 4 1 1 a2 vi Ds Y

27

i

p

where v0 e1i 1 1 a2i am0vi is the quiver velocity of plasma particles owing to the electric field of the incident electromagnetic wave.

If L L30 Â 1030ergs21 is the luminosity of pump radiation with frequency vi 2pni n10 Â 1010 rad s21 at a distance r

r8 Â 108 cm from the source then

e 2L 1a2

L1a2

v0 m0vi r2c

4X3 Â 109 30 cm s21X n10r8

28

When the phase matching conditions are met, the instability becomes more efficient, and Ds 2viv 2 v2 2 2c2ki´k 1 c2k2 < 0 represents

the dispersion relation of the Stokes mode. The thermal speed vt of the plasma can be expressed in terms of its energy spread in the laboratory frame. Let v^ c 1 2 1ag2^1a2 be

the velocity of the electron±positron plasma; then the velocity spread d v^ is given by (Hasegawa 1978; Gangadhara & Krishan 1992)

dv^ < c dgg3^ for g @ 1X 29

Now, using the Lorentz transformation of velocity, we can show that

dv^ dvz

dvz H

< dvz H vt

30

g2^ 1 1 v^vz Hac2 g2 g2

because vz H 0. Hence the thermal speed, in the plasma frame, is given by

vt c dg^ X

31

g^

For g^ 103, we get vt 3 Â 107dg^ cm s21. If we separate equation (27) into real and imaginary parts, we get two coupled simultaneous equations. By solving them numerically,

we find the growth rate G of stimulated Raman scattering. Fig. 2 shows the growth rate as a function of r8 and vi/vp. Since the plasma density decreases with r as 1/r3, the growth rate decreases as r8 increases. Also, if the radiation frequency becomes higher than the plasma frequency, then radiation and plasma do not couple resonantly, which leads to the decrease of the growth rate at higher values of vi/vp.

To study the variation of G in the a i±d g^ plane, we have made a contour plot (Fig. 3); the labels on the contours indicate the values of log (G). The Debye length increases with the increase of d g^, therefore the plasma wave experiences a large Landau damping, which leads to the drop in growth rate. The parameter a i is the ratio of amplitudes of electromagnetic waves that are polarized in the directions parallel and perpendicular to the ki±B plane. When a i is small, the wave polarized in the direction parallel to the ki±B plane becomes strong and

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Polarization of pulsar radio emission 835

will have a component along B. Therefore the coupling between radiation and plasma will be strong, which leads to the higher growth rate

at the smaller values of ai. For v ! c2ki´kavi, Stokes mode becomes resonant, and the anti-Stokes mode becomes non-resonant. Then equation (27) can be

written as

v 2 vl 1 iGl v 2 vl 1 iGs h 1 m 21018 L30 ma 1 a2i h rad2 s22Y

32

a

r78a2n10 1 1 a2i

where v2l v2p 1 3a2k2v2t ,

p

!

Gl p vp exp 2 1 2 3 1 nc

33

2 klD3

2 klD2 2

is the damping rate of the plasma wave, and the Debye length lD vtap2vp. For klD , 1, we find Gl < 2X5 Â 109ar38a2s21. The pair plasma collision frequency nc < 2X5 Â 1023 ln Lar38dg3^ s21 and the Coulomb logarithm ln L < 10. The collisional damping rate of the scattered electromagnetic wave is given by Gs v2pnca2v2s < 0X06an210r68dg3^s21.

For klD . 1, plasma waves are highly damped and hence the plasma loses its collective behaviour. Therefore stimulated Raman

scattering changes into induced Compton scattering of electromagnetic waves by the plasma particles. In the conventional treatment of

induced Compton scattering in pulsars (e.g. Blandford & Scharlemann 1976; Sincell & Krolik 1992) the collective effects of the plasma are

ignored. The collective treatment of the wave scattering by plasma particles is justified if the condition klD ! 1 or vph @ vt is met, where

vph wpak is the phase velocity of the plasma wave. This condition implies that the wavenumber of oscillation of the electrons in the beat

wave of the incident and scattered waves is much less than the inverse of the Debye length. It is not satisfied when the beat wave is strongly

Landau damped (klD . 1) or the beat wave will not feel the presence of a medium ([email protected]), so that the scattering process will be described

as induced Compton scattering.

When klD ! 1, by setting v vl 1 iG, we can solve equation (32) for the growth rate:

s

G 2 1 Gl 1 Gs ^ 1 Gl ÿ Gs2 1 4X3 Â 1018 L30 ma 1 a2i h X

34

2

2

r78a2n10 1 1 a2i h 1 ma

Stimulated Raman scattering is a threshold process: if the intensity of the pump exceeds the threshold, only then will it start converting its

energy into decay waves. The threshold condition for the excitation of stimulated Raman scattering is given by

L30

219 3a2 1 1 a2i h 1 ma

r2 9X3 Â 10 n10r8 GlGs m 1 a2h X

35

8 thr

a

i

The typical threshold intensities for stimulated Raman scattering are of the order of the observed intensities, implying that pulsar magnetosphere may be optically thick to Raman scattering of electromagnetic waves.

Figure 2. The growth rate G of stimulated Raman scattering is plotted with respect to r8 and v i/vp.

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Figure 3. A contour plot of G in the d^±a i plane. The labels on the contours indicate the log (G) values.

836 R. T. Gangadhara, H. Lesch and V. Krishan

The growth rate just above the threshold is given by

G 4X3 Â 108 L30 ma 1 a2i h s21Y

36

r28n10 1 1 a2i h 1 ma

which is proportional to L30. The maximum growth rate attainable for vp . G . Gl, on the other hand, is

s

G 109 L30

ma 1 a2i h s21X

37

n310r183a2 1 1 a2i h 1 ma

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3 STOKES PARAMETERS

When the phase matching conditions are satisfied, the growth rate G becomes large, and the scattered mode is amplified and becomes a normal mode of the plasma. Under some conditions, the scattered mode leaves the plasma with a polarization which may be different from the polarization of the incident wave. The polarization states of the incident and scattered waves can be described more accurately using the Stokes parameters (Rybicki & Lightman 1979):

Ij 1xj1Ãxj 1 1yj1ÃyjY

38

Qj 1xj1Ãxj 2 1yj1ÃyjY

39

Uj 21xj1Ãyj cos dj

40

and

Vj 21xj1Ãyj sin djY

41

where j i for the incident wave, and s for the scattered wave. The linear polarization is given by

q

L U2j 1 Q2j Y

42

and the polarization position angle is given by

xj 12 arctan UjaQjX 43

The transfer of energy between the modes will be efficient only when the energy of the pump wave is strong enough to overcome the damping losses or escape of the generated waves. Using the Manley±Rowe relation (Weiland & Wilhelmsson 1977)

Ii Is Y

44

vi vs

we find the relation between the incident flux Ii and the scattered flux Is:

v

Is 1 2 vi IiX

45

In the following two subsections, we consider the cases in which the polarization state of the incident wave is linear and circular, and compute the polarization states of the scattered mode.

3.1 Linearly polarized incident wave

Consider a linearly polarized electromagnetic wave (di 0, ai 10 and xi 908), which excites stimulated Raman scattering in the magnetospheric plasma. Using the plasma and magnetic field parameters discussed in the previous section, we compute the growth rate of stimulated Raman scattering. For klD ! 1, the instability becomes quite strong and the Landau damping of the Langmuir wave is minimal. Hence the stimulated Raman scattering is resonantly excited. Fig. 4 shows the behaviour of linear (solid line) and circular (broken line) polarization of the scattered mode with respect to h . It shows that, at some values of h which are close to 0.1, the linear polarization of the incident wave can be converted almost completely into circular polarization of the scattered wave. Charge density variations within the pulse window could cause the conversion efficiency of linear to circular polarization to vary.

The variation of the polarization angle of the scattered wave with respect to h , at different values of d , is shown in Fig. 5. For d 908 and h # 0X1, the scattered mode becomes orthogonally polarized with respect to the incident wave. Furthermore, if there is any variation in the plasma density or radiation intensity, the value of h fluctuates and the scattered modes produced with h # 0X1 become orthogonally polarized with respect to those produced with h . 0X1.

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3.2 Circularly polarized incident wave

Suppose that the incident wave is circularly polarized (di pa2 and ai 1); then the scattered mode will be linearly polarized for h # 0X2, while, at the other values, both linear and circular polarizations with different proportions will exist, as indicated by Fig. 6. For di , 0 and ds di 2 d , 0 the sense of circular polarization of the scattered mode becomes opposite to that of the incident wave. Hence, depending upon the plasma and radiation conditions, stimulated Raman scattering can change the polarization of the pulsar radio signals.

4 DISCUSSION

The variable nature of circular polarization is evident from the polarization distributions of Manchester, Taylor & Huguenin (1975), Backer & Rankin (1980)and Stinebring et al. (1984a,b). Very high degrees of circular polarization are occasionally observed in individual pulses, even up to 100 per cent (Cognard et al. 1996), while the integrated or average pulse profiles generally indicate a much smaller degree of circular polarization (e.g. Lyne, Smith & Graham 1971), which show that the sign of circular polarization is variable at any given pulse phase. Radhakrishnan & Rankin (1990) have identified two extreme types of circular polarization in the observations: (i) an antisymmetric type wherein the circular polarization changes sense in mid-pulse; and (ii) a symmetric type wherein it is predominantly of one sense. The correlation of sense of the antisymmetric type of circular polarization with the polarization angle swing indicates the geometric property of the emission process, and is highly suggestive of curvature radiation.

The diverse nature of circular polarization may be the consequence of the pulsar emission mechanism and the subsequent propagation effects in the pulsar magnetosphere (e.g. Melrose 1995; von Hoensbroech, Lesch & Kunzl 1998). It seems rather difficult to explain the various circular polarization behaviours using the widely accepted magnetic pole models (e.g. Radhakrishnan & Cooke 1969; Komesaroff 1970; Sturrock 1971; Ruderman & Sutherland 1975).

The symmetric type of circular polarization observed in some pulsars (e.g. PSR B1914113 and B0628±28), as shown in Figs 7 and 8, may be due to the propagation effects. However, it seems difficult for propagation effects to explain how the sign of the circular polarization can change precisely at the centre of the pulse in the case of the antisymmetric type, as seen in many pulsars, e.g. PSR B1859103 and B1933116 (Rankin, Stinebring & Weisberg 1989).

If we are to understand the radio emission mechanism, we must understand the physical state of the radio-loud plasma in the polar cap. It is this plasma that is the site of instabilities which are thought to produce coherent radio emission. The role of different propagation effects in the pulsar polarization has been discussed by Cheng & Ruderman (1979), Beskin, Gurevich & Istomin (1988) and Istomin (1992). The mechanisms proposed by these authors predict a frequency dependence for circular polarization, with weaker polarization at higher frequencies. This is seen in some pulsars (e.g. PSR B0835±41, B1749±28 and B1240±64), but it is not generally the case (Han et al. 1998). Istomin suggested that the linearly polarized incident wave becomes circularly polarized as a result of generalized Faraday rotation; however, it is observationally known that no generalized Faraday rotation is evident in pulsar magnetospheres (Cordes 1983; Lyne & Smith 1990).

We have presented a model to explain the polarization changes owing to the propagation of radio waves through the magnetospheric

Figure 4. The solid line and broken curves indicate the variation of linear (Ls) and circular (Vs) polarization of the scattered mode with respect to h. The normalizing parameter Is is the intensity of the scattered mode.

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Figure 5. The polarization angle xs of the scattered mode is plotted as a function of h, at different values of d (08, 308, 608 and 908).

838 R. T. Gangadhara, H. Lesch and V. Krishan

Figure 6. The variation of linear (Ls) and circular (Vs) polarization of the scattered mode, with respect to h, indicated by the solid line and broken curves, respectively. The normalizing parameter Is is the intensity of the scattered mode.

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Figure 7. PSR B0628±28, an example of a pulsar with `symmetric' circular polarization and high linear polarization (from Lyne & Manchester 1988).

Figure 8. The polarization of PSR B1914+13, a pulsar with strong circular polarization over the whole observed profile (from Rankin et al. 1989).

plasma. The features, like a large change in polarization angle, a sense reversal of circular polarization and extremely rapid temporal changes in intensity, would help us to explain many observations, for which the existing mechanisms have proved to be inadequate. Because of the very strong dependence of polarization angle on plasma parameters via the growth rate, in an inhomogeneous plasma medium the depolarization is a natural outcome. We believe that plasma processes such as the stimulated Raman scattering may be a potential mechanism for the polarization variability in pulsars. The circular polarization of a number of pulsars varies with frequency. The two clear

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examples are PSR B1240±64 and B2048±78, from which the opposite senses or transitions of circular polarization have been observed at lower and higher frequencies in the radio band (Han et al. 1998).

5 CONCLUSION

We have considered the stimulated Raman scattering of the transverse electromagnetic waves in the electron±positron plasma of a pulsar magnetosphere. Since the frequency of the incident wave is much higher than the plasma frequency in stimulated Raman scattering, the change in scattered wave frequency is very small compared with the incident wave frequency. The value of the radiation±plasma coupling parameter h is determined by the polarization of the incident wave, and its value can be determined only by non-linear analysis. In the process of three-wave interaction, the phase matching condition (equation 13) between the initial phases (d i, d s, d ) and the value of h determine the polarization state of the scattered wave.

Many short-time-scale polarization variabilities in individual pulses, such as variations in the amount of linear and circular polarization, a sense reversal of circular polarization, and polarization angle swings, can be accounted for by considering stimulated Raman scattering. The time-scales over which the changes take place in individual pulses are of the order of the e-folding time, which is the inverse of the growth rate of stimulated Raman scattering.

It seems rather unlikely that the diverse behaviour of circular polarization can be accounted for by a single mechanism. Both intrinsic emission and propagation effects seem possible. The strong symmetric type of circular polarization observed in some pulsars is most probably generated by propagation effects, such as stimulated Raman scattering. Further simultaneous observations over a wide frequency range would be valuable in sorting out the importance of propagation effects.

ACKNOWLEDGMENTS

We are grateful to Y. Gupta and A. von Hoensbroech for discussions and comments. Fig. 8 has been reproduced, with permission, from Astrophysical Journal, published by The University of Chicago Press. (q 1989 by the American Astronomical Society. All rights reserved.) We thank Dr. J. M. Rankin for allowing us to use Fig. 8.

REFERENCES

AsseÂo E., 1993, MNRAS, 264, 940 AsseÂo E., Pelletier G., Sol H., 1990, MNRAS, 247, 529 Backer D. C., Rankin J. M., 1980, ApJS, 42, 143 Barnard J. J., Arons J., 1986, ApJ, 302, 138 Beal J. H., 1990, in Brinkmann W., Fabian A. C., Giovannelli F., eds, Physical Processes in Hot Cosmic Plasmas. Kluwer, Dordrecht, p. 341 Benford G., 1992, ApJ, 391, L59 Beskin V. S., Gurevich A. V., Istomin Ya. N., 1988, Ap. C&SS, 146, 205 Blandford R. D., Scharlemann E. T., 1976, MNRAS, 174, 59 Blandford R. D., 1975, MNRAS, 170, 551 Cheng A. F., Ruderman M. A., 1979, ApJ, 229, 348 Cognard I., Shrauner J. A., Taylor J. H., Thorsett S. E., 1996, ApJ, 457, L81 Cordes J. M., 1979, Space Sci. Rev., 24, 567 Cordes J. M., 1983, in Burns M. L., Harding A. K., Ramaty R., eds, AIP Conf. Proc. 101, Positron±Electron pairs in Astrophysics. Am. Inst. Phys., New

York, p. 98 Drake J. F., Kaw P. K., Lee Y. C., Schmidt G., Liu C. S., Rosenbluth M. N., 1974, Phys. Fluids, 17, 778 Fried D., Conte S. D., 1961, The Plasma Dispersion Function. Academic Press, New York, p. 1 Gangadhara R. T., Krishan V., 1992, MNRAS, 256, 111 Gangadhara R. T., Krishan V., 1993, ApJ, 415, 505 Gangadhara R. T., Krishan V., 1995, ApJ, 440, 116 Gangadhara R. T., Krishan V., Shukla P. K., 1993, MNRAS, 262, 151 Han J. L., Manchester R. N., Xu R. X., Qiao G. J., 1998, MNRAS, 300, 373 Hasegawa A., 1978, Bell System Tech. J., 57, 3069 Istomin Ya. N., 1992, in Hankins T. H., Rankin J. M., Gil J. A., eds, Proc. IAU Colloq. 128, The magnetosphere structure and emission mechanism of radio

pulsars. Pedagogical Univ. Press, Zielona GoÂra, p. 375 Komesaroff M. M., 1970, Nat, 225, 612 Krishan V., Wiita P. J., 1990, MNRAS, 246, 597 Kruer W. L., 1988, The Physics of Laser±plasma interactions. Addison-Wesley, New York, p. 70 Lesch H., Gil J. A., Shukla P. K., 1994, Space Sci. Rev., 68, 349 Liu C.S., Kaw P.K. 1976, in Simon, A., Thompson W. B., eds, Advances in Plasma Physics. Interscience, New York, Vol. 6, p. 83 Lominadze Dzh. G., Machabeli G. Z., Usov V. V. 1983, ApS&S, 90, 19 Lyne A. G., Manchester R. N., 1988, MNRAS, 234, 477 Lyne A. G., Smith F. G., 1990, Pulsar Astronomy, Cambridge Univ. Press, Cambridge, 234 Lyne A. G., Smith F. G., Graham D. A., 1971, MNRAS, 153, 337 McKinnon 1997, ApJ, 475, 763 Manchester R. N., Taylor J. H., Huguenin G. R., 1975, ApJ, 196, 83

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