Long Period Pulsars

7/31/2019 Long Period Pulsars

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ON THE NATURE OF RADIO PULSARS WITH LONG PERIODS

D. Lomiashvili

Tbilisi State University, Chavchavadze Avenue 3, 0128, Tbilisi, Georgia; [email protected]

G. Machabeli

Abastumani Astrophysical Observatory, Al. Kazbegi Avenue 2a, 0160, Tbilisi, Georgia; [email protected]

and

I. Malov

Pushchino Radio Astronomy Observatory, P. N. Lebedev Institute of Physics, 142290, Pushchino, Moscow Region;

and Isaac Newton Institute of Chile, Pushchino Branch, Pushchino, Moscow Region, Russia; [email protected]

Received 2004 October 29; accepted 2005 September 29

ABSTRACT

We show that the drift waves near the light cylinder can cause modulation of the emission with periods of the orderof several seconds. These periods explain the intervals between successive pulses observed in magnetars and radiopulsars with long periods. The model under consideration makes it possible to calculate the real rotation periods of thehost neutron stars. They are less than 1 s for the investigated objects. The magnetic fields at the surface of the neutron

star are of the order of 1011

1013

G and equal to the usual fields for known radio pulsars.Subject headings: pulsars: individual (PSR J11196127, PSR J18141744, PSR J18470130,

PSR J21443933) stars: magnetic fields

1. INTRODUCTION

According to the plasma model, pulsar radio emission is gen-erated in the electron-positron plasma, which appears by the av-alanche process ofe+epair production. The accomplishment ofthis process requires fulfilment of the following condition:

BSchw ! Bs ! Bdl: 1

Here Bs 3:2 ;1019

(

PP)1/2

is the pulsar surface magnetic fieldinferred from observational values of the spin period and periodderivative, BSchw m

2e c

3/e f % 4:4 ;1013 G is the Schwingerlimit, i.e., the magnetic field at which the electron cyclotron en-ergy equals the electron rest-mass energy, and Bdl is the mag-netic field corresponding to the so-called death line (see Fig. 1;Young et al. 1999). The various configurations of the surfacemagnetic field correspond to different death lines. They dependnot only on the magnetic field configuration, but also signifi-cantly on whether the origin of the gamma quanta, which are re-sponsible for pair production, is curvature radiation or inverseCompton scattering ( Zhang & Harding 2000). As a mechanismof producing the gamma quanta, we took curvature radiation withthe sunspot configuration of the magnetic field. In this case, the

death line is defined by the condition that the potential dropacross the gap required to produce enough pairs per primary toscreen out the parallel electric field is larger than the maximumpotential drop available from the pulsar. The quantity Bdl can besolved from the following equation (Chen & Ruderman 1993):

7 logBdl 13 log P 78: 2

Recently, there were discovered three long-period radio pul-sars, PSR J21443933 (Young et al. 1999), PSR J18470130(McLaughlin et al. 2003), and PSR J18141744 (Camilo et al.2000; see Table 1), that break the condition above (the first onebreaks the right-hand side of the inequality, and the other twobreak the left). PSR J21443933 is distinguished by some other

characteristics. It has the lowest spin-down luminosity ( E

42IP/P3 % 3:2 ;1028 ergs s1) of any known pulsar. The beam-ing fraction (that is, the fraction of the celestial sphere sweptacross by the beam) is also the smallest, W % 1/300. On the otherhand, PSR J18470130 (McLaughlin et al. 2003) and PSRJ18141744 (Camilo et al. 2000) are isolated radio pulsars hav-ing the largest, magnetar-like, inferred surface dipole magneticfields yet seen in the population: 9:4 ;1013 and 5:5 ; 1013 G, re-spectively. These pulsars show apparently normal radio emissionin the regime of magnetic field strengthBs ! BSchw, where plasmamodels predict no emission should occur. However, the nature ofthe Schwinger limit is not clear and is the subject of long-termdebate. Baring & Harding (2001) proposed that for extremelystrong fields, photon splitting dominates over pair production andthe corresponding death line is a function of both field strengthandperiod. This is definitely neither a certain nora hard limit. Fur-thermore, Usov (2002) has argued that since only one photonpolarization mode splits, the onset of photon splitting does notsuppress the production of pairs. It is very important to investigatethis interesting problem, but it is beyond the framework of ourmodel, especially since we argue that the magnetic fields of neu-tron stars in our investigation are less than the Schwinger limit.

A model that explains the phenomenon of radio emission fromthese pulsars and all their anomalous properties does not exist.An important feature of our model, which provides a natural ex-planation of most of the properties of these pulsars, is the presenceof very low frequency, nearly transverse drift waves propagatingacross the magnetic field and encircling the open field line regionof the pulsar magnetosphere (Kazbegi et al. 1991b, 1996). Thesewaves only periodically change the direction of the radio emissionand are not directly observable. We give a description of thismodel in xx 2, 3, and 4. Some estimates of the rotational and an-gular parameters of these pulsars are given in x 5. We discuss theobtained results in xx 6 and 7.

2. EMISSION MODEL

As mentioned above, the pulsar magnetosphere is filled with

a dense relativistic electron-positron plasma. The e+

e

pairs1010

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# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.


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are generated as a consequence of the avalanche process (firstdescribed by Sturrock1971), andthey flow along the open mag-

netic field lines. The plasma is multicomponent, with a one-dimensional distribution function (see Arons 1981a, Fig. 1),containing the following:

1. electrons and positrons of the bulk of the plasma with amean Lorentz factor ofp and density np,

2. particles of the high-energy tail of the distribution func-tion with t and nt, stretched in the direction of positivemomentum,

3. an ultrarelativistic (b $ 106) beam of primary particles

with so-called Goldreich-Julian density (Goldreich & Julian 1969)nb % 7 ; 10

2BsP1(R0/R)

3 cm3 (where P is the pulsar period,R0 % 10

6 cm is the neutron star radius, Bs is the magnetic fieldvalue at the stellar surface, andR is the distance from the center ofthe neutron star). This density is much less than the density of thesecondary pairs np.

Let us note that such parameters are only possible under theassumption that strong and curved multipolar fields persist in thepair-production region, near the stellar surface (Arons 1981b;Machabeli & Usov 1989).

Such a distribution function should generate various wavemodes under certain conditions. The waves then propagate in thepair plasma of the pulsar magnetosphere, transform into vacuumelectromagnetic waves as the plasma density drops, enter theinterstellar medium, and reach an observer as pulsar radio emis-sion. These waves leave the magnetosphere propagating at rel-atively small angles to the pulsar magnetic field (Kazbegi et al.1991a).

Particles moving along the curved magnetic field undergo driftmotion transversely to the plane where the field line lies. The driftvelocity can be written as

u c!BRc

; 3

where !B eB/mc, B Bs(R0/R)3, Rc is the curvature radius of

the dipolar magnetic field line, is the relativistic Lorentz factorof a particle, and is the particle velocity along the magneticfield. Here and below the cylindrical coordinate system (x, r, )is chosen, with the x-axis directed transversely to the plane ofthe field line, while r and are the radial and azimuthal coor-dinates, respectively.

Generation of radio emission is possible if at least one of thefollowing resonance conditions is fulfilled:

cyclotron instability : ! k kxu !Bres

; 4

Cherenkov instability : ! k kxu 0: 5

These conditions are very sensitive to the parameters of the mag-netospheric plasma, particularly to the value of the drift velocity(see eq. [3]), and hence to the curvature of the magnetic field lines

(Kazbegi et al. 1996).It should be noted that in the absence of drift motion, the or-

dinary Cherenkov interaction implies that the phase velocity of awave equals the velocity of the particles in both value and direc-tion. In other words, an observer moving with the same velocityas the particles should detect the same phase of the wave for asufficiently long time ( $ 1 / ). This is, however, impossiblefor a wave propagating transversely to the particles velocity. Onthe other hand, the drift velocity (eq. [3]) is directed along thephase velocity of such a wave. This allows wave-particle reso-nant interaction.

3. GENERATION OF DRIFT WAVES

It has been shown (Kazbegi et al. 1991b, 1996) that in addi-tion to the waves mentioned above (the characteristic frequen-cies of which fall into the radio band) propagating with smallangles to the magnetic field lines, very low frequency, nearlytransverse drift waves can be excited. They propagate across themagnetic field, so that the angle between kand B is close to /2.In other words, k?/k31, where k? (k

2r k

2)

1/2. Assuming(!/!B)T1, (u/c)2T1, k/kxT1, and kr 0, we can writethe general dispersion equation of the drift waves in the follow-ing form (Kazbegi et al. 1991a, 1991b, 1996):

1 X

!2!

Zu2

c

1

! k kxua

@f@p

dp k2c

2

!2

!

; 1 X

!2!

Z=c

! k kxua

@f@p

dpk2c2

!2

!

kxkc

2

!2X

!2!

Zu=c

! k kxua

@f@p

dp

!2 0; 6

where denotes the sort of particles (electrons or positrons),!2 4ne

2/m, f is the distribution function, and p is the mo-mentum of the plasma particles.

Let us assume that

! k kxub i; 7

Fig. 1.Line A: Death line when curvature radiation with the sunspotconfiguration of the magnetic field is taken as the mechanism of producing thegamma quanta. Line B: B BSchw.

TABLE 1

Radio Pulsars with Long Periods

Pulsar No.

P

(s)

P

(1015 s s1)

Bs(1012 G)

E

(1032 ergs s1)

PSR J21443933 ....... I 8.5 0.48 2 0.00032PSR J18470130 ....... II 6.7 1275 94 1.7

PSR J18141744 ....... III 4.0 743 55 4.7

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where ub is the drift velocity of the beam particles (see eq. [3]).In the approximation kTkxub and k2xT!

2p/

3pc

2, the imag-inary part can be written as

Im ! %nb

np

1=2 3pb

!1=2kxub: 8

According to equation (7), the frequency of a drift wave can bewritten as

!0 Re ! k kxub % kxub: 9

Drift waves propagate across the magnetic field and encircle theregion of the open field lines of the pulsar magnetosphere. Theydraw energy from the longitudinal motionof the beam particles,as in the case of the ordinary Cherenkov wave-particle interac-tion. However, they are excited only if kxub 6 0, i.e., in thepresence of drift motion of the beam particles. Note that theselow-frequency waves are nearly transverse, with the electricvector being directed almost along the local magnetic field. Letus note that although kTkxub for the drift waves, there stillexists a nonzero k. It appears that the growth rate (eq. [8]) israther small. However, the drift waves propagate nearly trans-versely to the magnetic field, encircling the magnetosphere, andstay in the resonance region for a substantial period of time.Although the particles give a small fraction of their energy to thewaves and then leave the interaction region, they are continuouslyreplaced by the new particles entering this region. The wavesleave the resonance region considerably more slowly than the par-ticles. Hence, there is insufficient time for the inverse action of thewaves on the particles. The accumulation of energy in the wavesoccurs without quasi-linear saturation. The amplitude of the wavesgrows until the nonlinear processes redistribute the energy overthe spectrum. As was demonstrated by Kazbegi et al. (1991b), thestrongest nonlinear process in this case is the induced scattering of

waves on plasma particles. Therefore, the growth of the drift-waveamplitude continues until the decrement of the nonlinear wavesNL becomes equal to the linear decrement. As a result, oneobtains quasi-regular configurations of drift waves. Generally,the nonlinear scattering pumps the wave energy into the long-wavelength domain of the spectrum:

kmax % rLC cP

2: 10

Here rLC is the radius of the light cylinder.According to equations (9), (10), and (3), the period of the

drift waves can be written as

Pdr 2!dr

2kxub

kub

e42mc

BP2

b: 11

It appears that the period of the drift wave is of the order ofseveral seconds. It is possible to determine the relationshipbetween Pdr, the derivative, and the rate of slowing down ofthe neutron star from equation (11),

Pdr eB

22mcbPP: 12

For the considered values of the parameters, we obtain Pdr %10 P. This relation is kept during the entire life of the pulsar,until it stops emitting.

4. MECHANISM OF FIELD LINE CURVATURE CHANGE

Let us assume that a drift wave with the dispersion defined byequation (7) is excited at some place in the pulsar magnetosphere.It follows from theMaxwell equationsthatBr E(kxc/!); hence,Br3E for such a wave. Therefore, the excitation of a drift wavecauses particular growth of the r-component of the local magneticfield.

The field line curvature c 1/Rc is defined in a Cartesianframe of coordinates (x, y, z) (where the z-axis is directed per-pendicular to the plane of the field line) as

c 1 dy

dx

2" #3=2d2y

d x2; 13

where dy/dx By/Bx. Using :=B 0 and rewriting equa-tion (13) in cylindrical coordinates, we obtain

c 1

r

B

B

1

r

1

B

B2

B2@Br@

: 14

Here B B2 B2r

1/2 % B1 B2r/2B

2. Assuming that

kr31, we obtain from equation (14)

c 1

r1 kr

Br

B

: 15

From equation (15) it is clear that even a small change of Brcauses a significant change ofc. The variation of the field linecurvature can be estimated as

cc

% krBr

B: 16

It follows that even a drift wave with a modest amplitude Br $Br $ 0:01B alters the field line curvature substantially,

c/c $ 0:1. Since radio waves propagate along the localmagnetic field lines, curvature variation causes a change of theemission direction.

5. THE MODEL

There is an unequivocal correspondence between the observ-able intensity and (the angle between the observers line ofsight and the emission direction; see Fig. 2). The maximum ofthe intensity corresponds to the minimum of . The period of apulsar is the time interval between the neighboring maximums ofthe observable intensity (minimums of). According to this fact,we can say that the observable period is representative of thevalue of, and as it appears below, it might differ from the spinperiod of the pulsar:

cos A =K; 17

whereA and Kare unit guidevectors of the observers andemis-sion axes, respectively. In the spherical coordinate system (r, ,), combined with the plane of the pulsar rotation, these vectorscan be expressed as

A 1; 0; ; 18

K 1;t; ; 19

where 2/P is the angular velocity of the pulsar, is theangle between the rotation and observers axes, and is theangle between the rotation and emission axes (see Fig. 2).

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From equations (17), (18), and (19), it follows that

arccos sin sin cost cos cos : 20

In the absence of the drift wave, 0 constant, and con-sequently the period of is equal to 2/.

According to equation (16), in the presence of the drift wave,the fractional variation c/c is proportional to the magneticfield of the waveBr, which is periodically changing. So t is harmonically oscillating about 0 with an amplitude c/c and rate !dr 2/Pdr. Thus, we can write that

0 sin !drt : 21

According to equations (20) and (21), we obtain

arccos

sin sin 0 sin !drt cost cos cos 0 sin !drt

; 22

kmin 0 sin 2k!dr

; 23

where kmin is the minimum of afterkrevolutions of the pul-sar.1 The parameters of the pulse profile (e.g., width and height)significantly depend on what the minimal angle would be be-tween the emission axis and observers axis when the first onepasses the other (during one revolution). If the emission conedoes not cross the observers line of sight entirely (i.e., the min-imal angle between them is more than cone angle , see in

eq. [24a]), then we cannot observe the pulsar emission. On theother hand, inequality (24b) defines condition that is necessaryfor emission detection:

kmin > ; 24a

kmin < : 24b

Hence, for some values of the parameters , !dr, , , , ,and (set A), it is possible to accomplish the following regime:after every k m turn, the minimal value of (mmin ) satisfiescondition (24b), while for intervening values of k [1 k

(m 1), where k and m are positive integers], mmin satisfiescondition (24a). In that case, the observable periodPobs does notrepresent the real pulsar spin period but is divisible by it:

Pobs mP: 25

Hence,

Pobs m P; 26

where P is the pulsar spin period.The dipolar magnetic field strength on the neutron star surface

can be written as

B 3:2 ; 1019ffiffiffiffiffiffi

PPp

: 27

From equations (25), (26), and (27), it follows that

B Bobs

m: 28

After inserting equations (28) and (25) in equation (2), we obtain

7 logB 13 log P 7 log Bobs 13 log Pobs 6 log m ! 78:

29

Then

6 log m ! 78 7 log Bobs 13 log Pobs: 30

It can be verified that there exists a value for m that satisfiesequation (28) and the following condition simultaneously:

B Bobs

m< BSchw: 31

Thus, it is possible to fulfill the conditions necessary for e+e

pair production for some values ofm.There is a common problem for pulsars II and III presented in

Table 1 (their surface magnetic field strength exceeds the quan-

tum critical valueB > BSchw), whereas for PSR J21443933 the

Fig. 2.Geometry of rotation (), emission (K), and observers (A) axes.Angles and are constants, while and are oscillating with time.

1 The detection moment of any pulse is taken as the zero point of the time

reckoning.

Fig. 3.Oscillating behavior of with time for0 % 0:12, 0:12,!dr 2/17 s

1, 2/0:85 s1, and 0, with t1 2/ and t2 4/.

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problem appears in a different way. Equation (29) is not fulfilled.After substitutingBobs andPobs from Table 1 in equation (30), weobtain

6 log m ! 4: 32

Thus, if equation (32) is accomplished, it is possible for PSRJ21443933 to generate radio emission.

For better estimate of m, we can use observational data forbeaming fractions. From Figure 2, it appears that the pulse widthcan be expressed as

W P 2 sin 2 sin

; 33

after inserting equation (25) in equation (33), we obtain

mW

Pobs

sin

sin : 34

As was mentioned above, to accomplish the described regime(eq. [25]), the angle between the observers line of sight and theemission direction after one revolution from that moment whenthey were coincident ( 0; Fig. 3), 1min, must exceed . Since

1

min

2P

Pdr ; 35

if we assume that0 , then we getPdr 2Pobs 2mPand

max sin

m

; 36

if we substitute this equation in equation (34), we obtain

W

Pobs

sin =m

m sin : 37

Here the left-hand side is known from the observations. Equa-tion (37) gives us the ability to estimate the angular parametersof pulsars for given values ofm.

If we consider these pulsars in the framework of our model,their parameters (e.g., angular and spin parameters) get newreal values, shown in Table 2. According to the obtainedresults, the considered pulsars are placed on a P-B diagram asshown in Figure 4. Thus, we developed the theoretical model ofpulsar emission, in the framework of which we explained allspecific features of pulsars presented in Table 1.

6. DISCUSSION

It should be noted that this model is applicable to the entirepopulation of pulsars, but the effects caused by drift waves aredifferent depending on the values of the parameters in set A. Inthe case of large ( > ), the most interesting effect is thelengthening of the observable period (see eq. [25]), which isaccomplished only whenPdris divisible byPto high accuracy. Itexplains the lack of such kinds of pulsars.

In the case of small ( < ), the observable period doesnot increase (except for 0 j j % ), but some other interest-ing effects appear, such as drifting subpulses (Kazbegi et al. 1991c)and period and the period derivative oscillation phenomenon,which is observed in PSR B182811 (Stairs et al. 2000) andPSR B164203 (Shabanova et al. 2001). Some authors (Jones &

Andersson 2001; Link & Epstein 2001; Rezania 2003) have pro-posed different models to explain this phenomenon within theframework of free precession of the neutron star. As shown byShaham (1977) and Sedrakian et al. (1999), the existence of pre-cession in the neutron star is in strong conflict with the superfluidmodels for the neutron star interior structure. Therefore, we candeclare thatthere doesnot yet exist a self-consistent explanation ofthis fact. We plan to study this problem in detail in a forthcomingpaper.

IfPdris not divisible by P, then the observed intensity must bemodulated with the period of the drift wave. It is impossible toget such variations with integrated pulse profiles. Deviations ofintegrable pulse intensities damp each other. The only possibleway to prove this scenario is by single pulse observations. Such

observations really show intensity variations ( Karastergiou et al.2001). Although they do not have a harmonic nature (this is dueto various noises and insufficient resolution), it benefits our model.So if experiments are modified to evolve the oscillating com-ponent, it would help validate our theory.

Let us consider pulsars with very short periods. As mentionedabove, drift waves arise in the vicinity of the light cylinder. Theshorter the pulsar spin period, the smaller is the radius of the lightcylinder, and consequently, the larger is the magnetic field valuein the wave generation region [B % B Bs(R0/R)

3]. Thus, ifwe take this consideration into account, from equation (16) itfollows that for pulsars whose period is much less than 0.1 s, theamplitude of the oscillation in the emission direction would be sosmall ( < 1) that the presence of drift waves would not

cause any significant effect.

TABLE 2

RealValues of Pulsar Parameters

Pulsar m

Pdr(s)

P

(s)

P

(1015 s s1)

Bs(1012 G)

E

(1032 ergs s1)

(deg)

0 %

(deg)

(deg) W10 /P

PSR J21443933 ...... 10 17.0 0.85 0.048 0.2 0.032 7 7 1.5 0.1PSR J18470130 ...... 6 13.4 1.12 210 16 61 5 5 3 0.3

PSR J18141744 ...... 8 8.0 0.5 93 6.9 300 5 5 2 0.2

Fig. 4.P-Bs diagram of the real positions of the considered pulsars,

corresponding to Table 2.

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We believe that if the scenario defined above (enlarging of theobservable period due to drift waves changing the emission direc-tion) is accomplished, then there must be simultaneous modula-tion of the radio and X-ray emission. Some evidence of this factis detected radio emission from SGR 1900+14 (Shitov 1999) andthe anomalous X-ray pulsar 1E 2259+586 (Malofeev & Malov2001). Higher radio frequency emission from these magnetars hasnot been observed yet, but discussion about this issue differs fromthe objectives of our paper. However, detection of both types ofemissionis a rare event: because they are generated on differental-titudes, radio and X-ray emission propagate in different directions.This implies that the pulsars considered in this paper do not showX-ray emission.

7. CONCLUSIONS

After these considerations, we can divide radio pulsars into thefollowing groups, which are listed with their main requirements:

1. Rapidly rotating pulsars, for which is too small.None of the mentioned effects should exist for these.

2. Pulsars with < and(Pdr P)/PdrT1.In thiscase,period, period derivative, and pulse shape oscillation should ap-pear. In the case of low accuracy of equality between Pdr and P,subpulsedrift can be observed( Kazbegiet al. 1991c; Gogoberidzeet al. 2005).

3. Pulsars with < .These should show observed in-tensity variations, modulated with the period of the drift waves.

4. Pulsars with

> and (Pdr mP)/PT

1 (where m isapositive integer).These appear different from the real, long,observable rotation period.

Thus, long-period radio pulsars represent one of the branches ofusual pulsars and must be considered in the frameworks of tra-ditional theories for the specific values of the parameters in set A.

We thank the referee for helpful suggestions. This work waspartly supported by grants from the Georgian Academy of Sci-ence, the Russian Foundation for Basic Research (project 03-02-16509), and the NSF (project 00-98685).

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