Toward explanation of the X-ray - radio correlation in the Vela pulsar

0.3 MB·English

by  S. A. Petrova

Mon.Not.R.Astron.Soc.000,1–15(2009) Printed21January2009 (MNLATEXstylefilev2.2)
Toward explanation of the X-ray – radio correlation in the
Vela pulsar
9 S. A. Petrova ⋆
0
Institute of Radio Astronomy, NASof Ukraine, 4, Chervonopraporna Str., 61002 Kharkov, Ukraine
0
2

n
Received...
a
J

5 ABSTRACT

RecentobservationsoftheVelapulsarhaverevealedapeculiarconnectionofitsemis-
]
E sion in the soft X-ray and radio ranges. We suggest the model of the radio pulse
formation in the Vela pulsar, develop the theory of the radio photon reprocessing to
H
high energies and on this basis interpret the observed X-ray – radio connection. The
.
h processesofspontaneousandinducedscatteringofradiowavesoffthe spiralingparti-
p cles and their observationalconsequences are examined. The particles are assumedto
- acquire relativistic gyrationenergies due to resonantabsorptionof the radio emission
o
in the outer magnetosphere of a pulsar. The spectral and angular distributions of the
r
t spontaneously scattered power are analyzed and compared with the characteristics
s
of the particle synchrotron emission. The consequences of intensity transfer from the
a
[ radio beam to the backgroundin the course of induced scattering are studied as well.

It is demonstrated that the induced scattering can account for the basic features of
1
theVela’sradioprofileanditspulse-to-pulsefluctuations.Inparticular,itcanexplain
v
a greater role of the leading component and its earlier arrival in stronger pulses. The
2
studiesoftheradiophotonreprocessingtohighenergiesinapplicationtotheVelapul-
6
4 sar shows that the scatteredand synchrotronspectra peak at ∼0.8 keV and 0.2 keV,
29 −1 31 −1
0 respectively,withthe correspondingluminositiesof10 ergs and10 ergs .The
. reprocessing inside the light cylinder results in the profile component which stretches
1 fromtheradiopulsepositionto∼30◦earlierinpulsephase.Thesynchrotronemission
0
beyondthelightcylinderpresumablycontributestothecomponentwhichlagsthe ra-
9
dio pulse by ∼ 90◦. Within the framework of our model, the observed X-ray – radio
0
: connectionisexplainedintermsoftheinterplaybetweentheprocessesofinducedand
v
spontaneous scattering of the radio pulse.
i
X
Key words: pulsars: general – pulsars: individual (the Vela pulsar) – radiation
r mechanisms: non-thermal – scattering.
a
1 INTRODUCTION isthemain X-raypeak;besidesthat,duringtheweakradio
pulses(whicharrivelater)theexcessX-rayfluxshiftstothe
The radio emission of pulsars and their non-thermal high-
’trough’,whichfollowsthesecondX-raypeakandspansthe
energy emission have essentially different properties. They
longitudinal interval close to theradio pulse window.
are undoubtedly generated by the distinct mechanisms,
which are still a matter of debate (see, e.g., Harding et al. Mostofthehigh-energyemissionmechanismsdiscussed
2008a;Melrose2004,forareview).Atthesametime,there- intheliteraturedonotdependontheradiopulseintensity.
centobservations(Shearer et al.2003;Lommen et al.2007) Oneofthemechanisms,however,suggeststheradiophoton
have demonstrated that the high-energy profiles can be in- reprocessingintothehigh-energybandandthereforeimplies
fluenced by the radio pulse intensity. In the Crab pulsar, a physical connection between the radio and high-energy
theoptical pulsescoincident with thegiant radiopulsesare emissions of pulsars (Lyubarskii& Petrova 1998; Gil et al.
3%brighterthantheaveragepulse(Sheareret al.2003).In 2001;Petrova2003;Harding et al.2005,2008b).Intheouter
theVelapulsar,theshapeoftheX-rayprofilechangeswith magnetosphere of a pulsar, the radio emission is subject to
theradiopulseintensity(Lommen et al.2007):thestronger resonantabsorptionbythesecondaryplasmaparticles.Asa
theradiopulse(theearlieritarrives),themorepronounced result,theabsorbingparticlesacquiresubstantialtransverse
momenta (Lyubarskii & Petrova 1998). In case of strong
enough absorption, the longitudinal momenta can also in-
⋆ E-mail:[email protected] crease significantly (Petrova 2002). Then the spontaneous
2 S. A. Petrova
synchrotron radiation of the absorbing particles falls into whichcoincideswiththeradiopulseandextendstotheear-
theopticalandX-rayrange.Thishasbeensuggestedtoac- lier pulse phases. In thesoft X-rayrange, thecomponent is
countforthenon-thermalhigh-energyemissionoftheyoung alsopresent,andonecandiscernitstwopeaks,oneofwhich
andmiddle-agedpulsars(Petrova2003).Lateronthemodel (peak 3) precedestheradio pulseand another one (peak 4)
wasextendedtothemillisecondpulsarsbyincludingthein- coincides with it (Harding et al. 2002). The component is
fluenceofthenon-compensatedacceleratingelectricfieldon most pronounced at a few tenthskeV and vanishesabove a
theparticle momentum evolution (Hardinget al. 2005). few keV. In the observations of Lommen et al. (2007), the
Higher intensities of the optical pulses observed in the X-ray profile is integrated over the range of 2–16 keV, and
Crab pulsar during giant radio pulses (Shearer et al. 2003) this component looks as a trough, but still interacts with
maywellbeexplainedintermsofsynchrotronre-emissionof theradio pulse.
the absorbing particles. Resonant absorption of giant radio In thepresent paper, we explain thetrough at keV en-
pulsesshould lead toamorepronouncedmomentumevolu- ergiesanditsconnectiontotheradiopulse.Forthispurpose
tionoftheparticlesand,correspondingly,tothelargersyn- weturntoonemoremechanism–thespontaneousscattering
chrotronpoweremitted.Atthesametime,theintensityre- offthespiralingparticles,–whichdepositstheradiophotons
distributioninthehigh-energyprofileoftheVelapulsarand into the high-energy range. The radio photon reprocessing
its dependence on the radio pulse intensity (Lommen et al. to high energies implies a physical connection between the
2007)stillremainspuzzling.Thisproblemisaddressedinthe radio and high-energy emissions, which can manifest itself
presentpaper.Firstly,weexplaintheradioprofilestructure in the simultaneous fluctuations in these ranges. The fluc-
of the Vela pulsar and its intensity-dependent variations. tuations are believed to result from the variations of the
Secondly,we consider one more mechanism, which deposits physicalconditionsinthemagnetosphere.Inourmodel,the
the radio photons into the high-energy band, and interpret radiopulseparticipatesinboththespontaneousandinduced
the high-energy profile. Finally, the observational manifes- scatterings,anditwill beshownthattheinterplaybetween
tations of theX-ray – radio connection are analyzed. these processes in the course of the pulse-to-pulse fluctua-
The radio observations of the Vela pulsar tions of theplasma parameters can account for theX-ray –
(Krishnamohan & Downs 1983) have revealed that its radio connection observed in theVela pulsar.
profilestructuredependson thepulseintensity:in stronger Theplanofthepaperisasfollows.Section2isdevoted
pulses,theleadingcomponent(theprecursor)ismuchmore tothetheoryofspontaneousscatteringoffthespiralingpar-
pronounced and arrives somewhat earlier, whereas the rest ticles. The spectral and angular distributions of the scat-
of thepulse(themain pulse) is markedly weaker an retains tered power are examined and compared with those of the
its position in the pulse window. Krishnamohan & Downs synchrotronradiationofthescatteringparticles.Thedetails
(1983) have suggested that the precursor component ofinducedscatteringoftheradiowavesbelowtheresonance
originates in a separate emission region, at higher altitudes are given in Sect. 3. In Sect. 4 we apply our formalism to
in the magnetosphere. Then its location in the pulse theVelapulsar.Theradioprofileformation isconsideredin
profile can be explained in terms of rotational aberration. Sect.4.1,thehigh-energyemission isaddressedinSect.4.2,
However, if the emission regions of the precursor and the and the observational manifestations of the X-ray – radio
main pulse are completely independent, it is difficult to connectionareinvestigatedinSect.4.3.Ourresultsaredis-
understand the observed intensity redistribution between cussed and summarized in Sect.5.
thesecomponents,whichiscontrolled bythetotalintensity
of the pulse.
We suggest that the precursor component results from
2 SPONTANEOUS SCATTERING OFF THE
theinducedscatteringofthemainpulseintothebackground
SPIRALING PARTICLES
(see also Petrova 2008b). The main pulse emission is scat-
teredofftheparticles,whichparticipateintheresonantab- The cross-section for the magnetized scattering by the par-
sorption and, consequently, perform relativistic helical mo- ticle at rest was first obtained in Canuto et al. (1971).
tion. The induced scattering transfers the radio intensity In application to the pulsar magnetosphere, the scatter-
from the main pulse to theprecursor component. It will be ing in a strong magnetic field by the particles stream-
shown that stronger inducedscattering entails larger inten- ing relativistically along the magnetic lines was examined
sitiesoftheresultantradiopulseandalsoitsearlierarrival. in Blandford & Scharlemann (1976); Lyubarskii & Petrova
Thus,itistheprocessthatcanaccountforthetheradiopro- (1996). Wilson & Rees (1978) have considered the non-
file structureof the Vela pulsar and its intensity-dependent magnetized induced scattering in the pulsar wind.
variations. Let us consider the radio wave scattering off the par-
The high-energy profile of the Vela pulsar has the ticles performing relativistic helical motion in themagnetic
following structure (Kanbach et al. 1994; Gouiffes 1998; field of a pulsar. Deep in the magnetosphere the particles
Strickman et al. 1999; Harding et al. 2002; Sanwal et al. stream relativistically along theopenmagneticlines. Inthe
2002; Shibanov et al. 2003; Romani et al. 2005). There are vicinityoftheradioemissionregion,themagneticfieldisso
two main peaksseparated by about a half of thepulsar pe- strong that any perpendicular momentum of the particles
riod. None of them coincide with the radio pulse, and peak is almost immediately lost via synchrotron re-emission and
1 lags it by 90◦ in pulse phase. In the optical and soft the radio wave frequency in the particle rest frame is much
∼
X-ray profiles, peak 1 noticeably shifts with frequency to- less than the electron gyrofrequency, ωηγ ω eB/mc
G
≪ ≡
wardtheradiopulseposition,whereasathigherenergiesthe (hereη 1 βcosθ,βistheparticlevelocityinunitsofc,θ
≡ −
components keep their positions fixed. Gouiffes (1998) has isthewavevectortilt tothemagnetic field,γ istheparticle
discovered an additional component of the optical profile, Lorentz-factor, γ (1 β2)−1/2).
≡ −
X-ray – radio correlation in the Vela pulsar 3
As the magnetic field strength rapidly decreases with sinθ ηη′ β2(1 cosθcosθ′)/2 +η2sinθ′cos∆φ 2,
distance from theneutron star, B r−3, in the outermag- × − ⊥ −
netosphere the radio waves pass th∝rough the cyclotron res- dσ(cid:8)sBB =(cid:2) re2 Ω2 s4J′2 sβ⊥sinθ(cid:3)′ β⊥2 (cid:9)
onance, ωηγ = ω , where they are subject to resonant ab- dO′ γ2η′5 ω2η2 s η′ η2
G (cid:18) (cid:19)
sorption.Asaresultofthisprocess,theincidentradioemis-
sion is partially absorbed and the particles acquire trans- × sinθ ηη′−β⊥2(1−cosθcosθ′)/2 +η2sinθ′cos∆φ 2,(2)
versemomenta.Intheresonanceregion,themagneticfieldis
wh(cid:8)ere re(cid:2)is the classical electron rad(cid:3)ius, βk and β⊥ are(cid:9)the
weak enough, and thespontaneous synchrotron re-emission
components of the particle velocity parallel and perpendic-
does not prevent the momentum growth. As is shown in
ular to the ambient magnetic field, respectively, in units of
Petrova (2002, 2003), the particle gyration becomes rela- c, ∆φ = φ φ′ is the difference of the azimuthal wavevec-
tivisticattheverybottomoftheresonanceregion,andfur- −
tor components of the incident and scattered radiation and
ther on the transverse and total momenta of the particles
the superscripts of the cross-sections denote the initial and
continuegrowing.
finalpolarization statesofthewaves.MakinguseofEq.(1)
Pulsarradioemissionisessentiallybroadbandand,cor-
in Eq. (17) of Petrova (2008a) yields the zeroth-harmonic
respondingly, the resonance region is sufficiently extended.
cross-sections
Over most part of this region there is a significant amount
dσAA r2 sin2θsin2θ′
of the photons with frequencies well below the resonance, 0 = e ,
ωη Ω eB/γmc. We are interested in the scattering of dO′ γ2η′5 γ4
k
≪ ≡
the under-resonance radio emission off the relativistic spi-
dσAB r2 ω2η2
raling particles. In our case the incident radiation presents 0 = e
thetransverseelectromagneticwavespolarizedeitherinthe dO′ γ2η′3 Ω2
planeoftheambientmagneticfield(A-polarization)orper- β2 sinθsinθ′(cosθ′ β ) 2
pendicularly tothis plane (B-polarization). (cosθ β )cos∆φ ⊥ − k ,
× − k − 2η′2
The scattering by the electron on a circular orbit has (cid:20) (cid:21)
been examined in Petrova (2008a). It has been shown that dσBA r2 ω2η2
the under-resonance waves are predominantly scattered to dO0′ = γ2ηe′5 Ω2
high harmonics of theparticle gyrofrequency and may con-
tribute to the observed high-energy emission. (It has also β2 sinθsinθ′(cosθ β ) 2
(cosθ′ β )cos∆φ ⊥ − k ,
been shown that the effect of the scattering on the particle × − k − 2η2
momentaisnegligible.)Inthepresentpaper,weextendthis (cid:20) (cid:21)
formalism to the case of relativistic helical motion of the dσ0BB = re2 ω2η2 sin2∆φ. (3)
scattering particles, examine the spectral and angular dis- dO′ γ2η′3 Ω2
tributionsofthescatteredradiationandcomparethemwith
The total scattering cross-section has theform
those of the synchrotron radiation of thesame particles.
Given that the incident radio frequency is well below dσij dσij ∞ dσij
= 0 +2 s . (4)
the resonance, the components of the differential scatter- dO′ dO′ dO′
ing cross-sections at thenon-zero harmonics of the gyrofre- Xs=1
quency, s = 0, for different polarization channels are given Comparing Eqs. (2) and (3), one can see that the zeroth-
6
byEq.(16) inPetrova(2008a).Therelativistictransforma- harmonic term makes negligible contribution to the total
tion of the cross-section reads cross-section.
dσ dσ η2 The power scattered by an electron is written as
= , (1)
dO′ (cid:16)dO′(cid:17)c η′3γk2 Pij =2I ∞ 2π π dσsij sinθ′dθ′dφ′, (5)
where γk is the Lorentz-factor of the longitudinal motion, dO′
theprimesdenotethecharacteristics ofthescatteredradia- Xs=1Z0 Z0
tionandthesubscript’c’referstotheguiding-centreframe.
whereI =I(ω,θ,φ)istheincidentintensity.Itisinteresting
With Eq. (1), the cross-section components for the case of
tocomparethescatteredpowerwiththesynchrotronpower
relativistichelicalmotionofthescatteringparticlesarewrit-
of theelectron,
ten as
π
ddσOsA′A = γ2rηe2′5 ωΩ4η44s4Js2 sβ⊥ηs′inθ′ Psyn = e2cΩ2 ∞ s2 sinηθ′′3dθ′
(cid:18) (cid:19) s=1 Z
X 0
×ssiinn22θθ′ cosθ′−βk 4, cosθ′−βk 2J2 sβ⊥sinθ′ +β2J′2 sβ⊥sinθ′ ,
dσAB (cid:0) r2 Ω4 (cid:1) sβ sinθ′ ×"(cid:18) sinθ′ (cid:19) s (cid:18) η′ (cid:19) ⊥ s (cid:18) η′ (cid:19)#
s = e s4J′2 ⊥
dO′ γ2η′5 ω4η4 s η′ where the first and the second terms correspond
(cid:18) (cid:19)
to the A- and B-polarizations, respectively. One can
×dσβsB⊥2Asin=2θ(cid:0)rce2osθΩ′−2 βsk4(cid:1)J22, sβ⊥sinθ′ cosθ′−βk 2 βcohbkr)so2e/trrvsoeinn2ctθah′sa]eJt.s2/H(βo∂⊥2wPJesisv′A2e,/r∂,(Oith′=e)/a(∂nAPg,usBilBa)/r,∂asOinm′d)ilsaprel=yctrtaol[(tdchoisestθrsi′bynu−--
dO′ γ2η′5 ω2η2 s η′ η2sin2θ′
(cid:18) (cid:19)(cid:0) (cid:1) tions of the scattered power are somewhat different.
4 S. A. Petrova
To examine these distributions in more detail we allow AppendixA,one can obtain the normalized spectral distri-
for the relativistic character of the electron gyration and butions
make use of the asymptotic representations of the Bessel
∞
function and its derivative with respect to the argument ξ 243√3
fAA(y)= y3 K (y) K (x)dx ,
at ξ s 0: 256π  4/3 − 5/3 
→ − Z
y
Js(ξ)= πε1√/23K1/3 3sε3/2 , 81√3 8 ∞
(cid:16) (cid:17) fAB(y)= y3 K (y)+K (y) K (x)dx ,
Js′(ξ)= π√ε3K2/3 3sε3/2 , 256π 3y 1/3 4/3 −Zy 5/3 
(cid:16) (cid:17)  
whereε 1 ξ2/s2 andK (x)isthemodified Bessel func- ∞
tion. It i≡s con−venientto intrµoduce thevariables fAB(y)= 8414√8π3y3 K5/3(x)dx−K2/3(y),
ψ= β0γ(cosθ′−βk) and y= 23sγ0−3, (6) Zy 
1 β2 ∞
− k 81√3
fBA(y)= y3 K (x)dx+K (y) ,
where βqis the normalized velocity of the electron in the 448π  5/3 2/3 
0 Z
guidingcentreframe andγ (1 β )−1/2.Theinvariance y
0 ≡ − 0  
of the transverse momentum implies that β0γ0 =β⊥γ and, ∞
correspondingly,γ0 =γ/γk.Inthecaseunderconsideration fA (y)= 9√3y K (x)dx K (y) ,
γ 1 and β 1. As the functions K (ξ) and K (ξ) syn 16π  5/3 − 2/3 
0 ≫ 0 ≈ 1/3 2/3 Z
aresignificantonlyfortheargumentsξ.1,onecanextend y
 
the limits of integration over ψ to ( , ). Furthermore, ∞
aresptlhaceesetfhuencsutimonmsapteioankoavtehrigshbhyairnmt−eog∞nriacst∞,iosn∼ovγe03r,yo.neThcaenn fsByn(y)= 91√6π3y K5/3(x)dx+K2/3(y), (9)
Z
thescattered power is written as y
 
r2 Ω4 81 where it is taken that ∞(fAA + fAB)dy = ∞(fBA +
PA=4πIγe2ω4η4γ06γ2sin2θ32π2 fBB)dy = 0∞(fsAyn + fRsB0yn)dy = 1. ProceedinRg0from the
approximation of the modified Bessel function at small ar-
∞ ∞ R
guments,
y4dy dψ ψ4(1+ψ2)K2 (ξ)+ψ2(1+ψ2)2K2 (ξ) ,
× 1/3 2/3 K (y) 2µ−1Γ(µ)y−µ, y 1,
Z Z µ
0 −∞ (cid:2) (cid:3) ≈ ≪
whereΓ(µ)isthegamma-function,onecanfindtheasymp-
r2 Ω2γ6γ4 sin2θ 81
PB =4πI e 0 1+ totic behaviourof thespectra at y 0:
γ2 ω2 2 2γ2η2 32π2 →
(cid:18) k (cid:19)
243√3 5
∞ ∞ fAA = 2−5/3Γ(1/3)y5/3, fAB = fAA,
256π 3
y4dy dψ ψ2(1+ψ2)K2 (ξ)+(1+ψ2)2K2 (ξ) ,(7)
× 1/3 2/3 81√3
Z Z fBA = 2−4/3Γ(2/3)y7/3, fBB =3fBA,
0 −∞ (cid:2) (cid:3) 448π
where the superscripts of P denote the polarization of the
9√3
incident waves, ξ (y/2)(1+ψ2)3/2, and the synchrotron fA = 2−4/3Γ(2/3)y1/3, fB =3fA , (10)
≡ syn 8π syn syn
power reads
Note that the scattered radiation has much steeper spec-
P = e2Ω2γ2γ02 9 tra. Provided that y 1, Kµ(y) π/2yexp( y) and
syn c 8π2 the spectra drop expo≫nentially. The≈overall spectra−l distri-
p
∞ ∞ butions(9)areplottedinFig.1.Onecanseethatthepower
y2dy dψ ψ2(1+ψ2)K2 (ξ)+(1+ψ2)2K2 (ξ) .(8) ofthescatteredradiation peaksatmarkedlylargerfrequen-
× 1/3 2/3 ciesthanthesynchrotronpowerinbothcasesoftheincident
Z Z
0 −∞ (cid:2) (cid:3) A- and B-polarizations (y 1.25 and 2.5, respectively,
peak
≈
Integrating Eqs. (7)-(8) over the angular coordinate ψ whereas the synchrotron peak lies at y 0.3).
≈
withthehelpofthewell-knownintegralsofthesynchrotron Making use of theformula
theory
∞
ρ+1+µ ρ+1 µ
∞ π ∞ Kµ(y)yρdy=2ρ−1Γ 2 Γ 2− ,
ψ2(1+ψ2)K12/3(ξ)dψ= √3y  K5/3(x)dx−K2/3(y), Z0 (cid:16) (cid:17) (cid:16) (cid:17)
Z Z
−∞ y ρ+1>µ,
 
∞ ∞ one can integrate the spectral distributions to obtain the
π
(1+ψ2)2K22/3(ξ)dψ= √3y  K5/3(x)dx+K2/3(y) total power in each polarization:
−Z∞ Zy πr2 Ω4 13
and the analogous integrals (A14) and (A18) calculatedin PAA= 2eIγ06ω4η4 sin2θ, PAB = 3 PAA,
X-ray – radio correlation in the Vela pulsar 5
2πr2 Ω2 sin2θ the course of induced scattering between the photon states
PBA = eIγ6γ2 1+ , PBB =13PBA,
3 0 ω2 2γ2η2 corresponding to different harmonics of the gyrofrequency
(cid:18) k (cid:19)
thephotonsaretransferredfromthehigherharmonicstothe
PA = e2Ω2γ2γ02, PB =7PA . (11) lowerones.Thus,theincidentunder-resonancewavessuffer
syn 12c syn syn only the zeroth-harmonic inducedscattering, ω′η′ =ωη.
It should be noted that the polarization states of the As is shown in Petrova (2008c), the induced scattering
scattered radiation in the cases of incident A- and B- fromseveralfirstharmonicstotheunder-resonancestatecan
polarizations are distinct and both differ from the syn- alsobenoticeable.Atthesametime,theinducedscattering
chrotron case. Besides that, R (PAA +PAB)/(PBA + from the high harmonics, s 1, is negligible. Firstly, the
PBB) (Ω2/ω2η2)(sin2θ/η2γ2≡). Although the quantity spectral intensities of the pu≫lsar high-energy emission are
∼
sin2θ/η2γ2γ2 may be much less than unity, in our consid- muchlessthantheradiointensities.Besidesthat,incontrast
k 0
eration R 1, since only the leading terms in Ω2/ω2η2 tothespontaneousscattering,theinducedscatteringchiefly
≫
areretained.Thus,thewavesoftheA-polarizationarescat- transfers the photons to the neighbouring harmonics, the
teredmuchmoreefficiently.Notealsothatthelinearization process becoming less efficient at s 1.
≫
techniqueapplied for thederivation of thescattering cross- Keeping in mind the above considerations, we are in-
section(2)isvalidonlyuntilthepowerscatteredislessthan terested in the under-resonance induced scattering, ω′η′ =
the synchrotron power of the particle (for more details see ωη Ω,off theparticles performingrelativistic helical mo-
≪
Petrova 2008a). tion.Itshouldbenotedthat,accordingtoEq.(3),thescat-
The angular distributions can be obtained by integrat- tering between the states with the A-polarization strongly
ing Eqs. (7)-(8) over y with thehelp of theintegral dominates that in the other polarization channels, since
only the cross-section dσAA/dO′ does not contain the fac-
∞
2ρ−3 ρ+ν+µ ρ+ν µ torω2η2/Ω2 1 . Hence,we dwell on thescattering in the
Kµ(x)Kν(x)xρ−1dx= Γ(ρ)Γ 2 Γ 2− channelA ≪A.
Z0 (cid:16) (cid:17) (cid:16) (cid:17) Pulsar→radioemissionisknowntobehighlydirectional.
ρ ν+µ ρ ν µ At any point of the emission cone it is concentrated into a
×Γ −2 Γ −2− , ρ>ν+µ>0, narrowbeamoftheopeningangle.1/γk,whichistypically
(cid:16) (cid:17) (cid:16) (cid:17) much less than the angular width of the emission cone. Far
and they takethe form
enough from the emission region, the radio emission prop-
315 11ψ4 315 13ψ2 agates quasi-transversely with respect to the ambient mag-
gAA= , gAB = ,
512(1+ψ2)13/2 512(1+ψ2)11/2 neticfield, 1/γ θ.1.
k ≪
Therateofinducedscatteringisdeterminedbythepar-
45 11ψ2 45 13
gBA= , gBB = , ticle recoil in the scattering act, and hence, the induced
512(1+ψ2)13/2 512(1+ψ2)11/2
scattering at large angles, out of the radio beam, may be
3 5ψ2 3 7 muchmoreefficientthanthescatteringinsidethebeam.At
gA = , gB = , (12)
syn 32(1+ψ2)7/2 syn 32(1+ψ2)5/2 thesametime,theformerprocessmaystartonlyifinitially
therearesomephotonsoutsidethebeam.Suchbackground
where ∞ (gAA + gAB)dψ = ∞ (gBA + gBB)dψ = photons may be present, e.g., due to the spontaneous scat-
−∞ −∞
∞ (gA + gB )dψ = 1. These distributions are plotted teringfromthebeam.Althoughthebackgroundphotonsare
−∞ syRn syn R
in Fig. 2. Note the drop of gAA +gAB at ψ = 0 and the veryfew,theycanstillstimulateefficientinducedscattering
R
narrowness of the peak of gBA+gBB as compared to the from the beam, and finally a significant part of the beam
synchrotron case. Integration of the angular distributions intensitymaybedepositedtothebackground(seeSect.4.1
using theformula below). At the beginning of induced scattering, the back-
ground occupation numbers grow exponentially, the expo-
∞
dψ Γ(1/2)Γ(ρ 1/2) nentbeingdependentonthephotonorientation.Hence,the
(1+ψ2)ρ = Γ(ρ)− beamphotonsarescatteredpredominantlyintothestate~k′
−Z∞ correspondingtothemaximumscatteringprobability.Thus,
the induced scattering from the radio beam into the back-
leads again to thetotal powers given byEq. (11).
ground gives rise to a narrow scattered component. As is
shown in Petrova (2008c), in the case considered it is di-
rected at the angle θ′ γ /γ to the ambient magnetic
3 INDUCED SCATTERING OFF THE max ∼ 0
field. The spectral intensities of the beam and the back-
SPIRALING PARTICLES
ground evolve as
The brightness temperatures of pulsar radio emission are
I/x I
so high that the induced scattering is believed to dominate Iν = 1+1/x, Iν′ = 1+1/x, (13)
the spontaneous one. Below we concentrate on the induced
scattering of the under-resonance radio waves, ωη ≪ Ω. whereI ≡Iν+Iν′ =constisthetotalintensityofthebeam
Asthemagneticfieldstrengthdecreaseswithdistancefrom and the background, x I(0)/I(0) exp(Γ), I(0) and I(0)
theneutron star, thewaveof agiven frequencysuccessfully ≡ ν′ ν ν ν′
are the initial intensities ofhthe beami and the background
passesthroughtheresonancesofincreasingly higherorders.
and
Theunder-resonancescatteringtakesplaceatthelowestal-
titudes,anditshouldbemostefficientbecauseoflargerinci- 12IN r2r
Γ= e e . (14)
dentintensityand plasmanumberdensityin thisregion. In mν′2γγ2
0
6 S. A. Petrova
The quantity Γ characterizes the scattering efficiency, where α is the spectral index of the pulsar radio emission,
whereas x the extent of intensity transfer from the radio ν 108 Hz,
0
≈
beamtothebackground.Aslongasx 1,thebackground
intensitygrowsexponentially,Iν′ ∼Iν(0′)≪exp(Γ),whereasthe Iν0 = SLν , (19)
beam intensity is almost unaltered, I I(0). Given that 0
ν ν
≈
x 1, the background intensity becomes comparable with L is the total radio luminosity of the pulsar, S = πr2ϑ2/4
th≫e initial radio beam intensity, Iν′ ≈ Iν(0), and enters the is the cross-section of the radio emission cone at a distance
stage of saturation, whereas the beam intensity noticeably r and ϑ is thepulse width in theangular measure. Then
dfuenccretiaosness,oIfνx∼arIeν(0s)h/oxw.nTihneFfiign.al3.inNteontesittihees nIνararonwdnIeνs′saosf Γ=900 L 0.1s B⋆ ϑ −2 r −4
the range of x, where both intensities are comparable in 1029ergs−1 P 1012G 0.1 108cm
(cid:16) (cid:17) (cid:16) (cid:17)
magnitude.
ν′ −2 ν −α κ 103 γ −2
0 . (20)
× 109Hz 108Hz 102 γ 10
(cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17)
4 APPLICATIONS TO THE VELA PULSAR
Taking L = 2 1029 ergs−1, ϑ = 0.15, r = 3 108 cm
× ×
4.1 Radio profile formation as well as P and B listed above, we obtain Γ = 30. Thus,
⋆
in theVela pulsar theinduced scattering of the radio beam
Let us examine the radio profile evolution of the Vela pul-
intothebackgroundcanindeedbeefficient.Furthermore,in
sar as a result of induced scattering off the spiraling parti-
the course of pulse-to-pulse fluctuations of the parameters,
cles.Thevariationsoftheconditionsinthescatteringregion
theinducedscatteringcanfromtimetotimeenterthestage
should result in the pulse-to-pulse fluctuations of the radio
of saturation, when theoriginal radio beam is substantially
profile, and it is the fluctuations that are believed to be
suppressed (see Fig. 3).
connected with those of thehigh-energy emission.
It should be kept in mind that the pulsar radio beam
Recall that our consideration is restricted to the in-
is broadband and its angle of incidence increases with dis-
duced scattering between the photon states with the A-
tancefromtheneutronstar,θ r(seebelow).Thereforeat
polarization.TheradioemissionoftheVelapulsarisknown ∝
different altitudes r the background component of a given
to have almost complete linear polarization. It is generally
frequency ν′ = νθ2(r)γ2/2γ2 is fed by the beam radiation
believed that in this pulsar only one polarization mode is 0
of different frequencies ν r−2. As at lower frequencies
present in any radio pulse and at any pulse phase. We as- ∝
the beam radiation is more intense, the induced scattering
sumethatthisistheA-mode(seeSect.5forfurtherdiscus-
is more efficient at higher altitudes above the neutron star
sion).
(formoredetailseePetrova2008b).Thus,theeffectivescat-
In order to conclude whether the induced scattering
tering region lies far from the emission region, at distances
may affect the Vela’s radio profile noticeably we start from
oftheorderofthecyclotronresonanceandthelightcylinder
estimating the level of the background radiation, which re-
radii, r r ,r .
sults from the spontaneous scattering of the radio pulse: ∼ c L
I(0)/I(0) =N r(ω/ω′)dσ/dO′.UsingEq.(3)andtakinginto The location of the scattered component in the pulse
aνc′counνt thateθ′ 1/γ and γ =γ/γ , we obtain profileispredominantlydeterminedbytheeffectofthemag-
≈ k k 0 netosphere rotation (Petrova 2008b). In the scattering re-
IIνν((00′)) ≈ 2Nγe02re2r. (15) lgoiocanl,mthaegninectiidcefinetldbdeairmectmioanke~bs.Nthoeteanthgalet∼inrth/e2rfLramwiethcotrhoe-
tating with theneutron star the beam is shifted in by r/r
L
Itisconvenienttonoramlizethenumberdensityofthescat-
oppositely to the direction of rotation and again makes the
tering particles bythe Goldreich-Julian density,
angle r/2r with the magnetic field (see Fig. 4). As for the
L
κB scattered component, in the corotating frame it is directed
N = , (16)
e Pce approximatelyalong themagneticfield.Hence,inthepulse
profile it precedes the original radio beam by ∆λ = r/2r
whereκisthemultiplicityfactoroftheplasmaandP isthe L
in phase. Thus, the induced scattering of the pulsar beam
pulsar period. With the dipolar geometry of the magnetic
field, B r−3, one can estimate Eq. (15) as into the background gives rise to the leading component of
∝ theprofile, which is usually called theprecursor.
Iν(0′) =10−110.1s B⋆ κ γ0 −2 r −2, (17) To analyze the radio profile structure of the Vela pul-
I(0) P 1012G102 10 108cm sar in more detail it is necessary to take into account the
ν
(cid:16) (cid:17) (cid:16) (cid:17) finitude of the pulse width. In the scattering region, differ-
where B is themagnetic field strength at the neutron star
⋆ ent radio beams (which constitute the emission cone and
surface and all the quantities are normalized to their char-
appear in the pulse profile at different pulse phases) make
acteristic values. For the parameters of the Vela pulsar,
somewhat differentangleswith thelocal magneticfield:the
P = 0.089 s and B = 3.4 1012 G, we have I(0)/I(0)
4 10−12.Hence,the⋆induced×scatteringbecomesνsi′gnifiνcan≈t angle of incidence θ ranges roughly from r/2rL −ϑ/4 to
× r/2rL+ϑ/4.Forafixedfrequencyoftheincidentradiation,
(i.e. x∼1) for thescattering efficiencies Γ≈30. ν,thescatteringefficiencyΓdependsonθ:Γ ν′−2 θ−4,
To estimate Γ we take that I ≡ Iν(0)+Iν(0′) ≈ Iν(0) and and therefore different parts of the main pu∝lse suff∝er the
present the radio beam intensity as scatteringofvariousstrength.Theangleθ enteringEq.(20)
ν −α corresponds to the corotating frame, and, as can be seen
I(0) =I , (18)
ν ν0 ν fromFig.4,smallervaluesofθ arecharacteristicoftherays
0
(cid:16) (cid:17)
X-ray – radio correlation in the Vela pulsar 7
formingthetrailingpartoftheprofile.Thus,itisthetrailing dominate at higher frequencies. This is proved by the ob-
part that should be scattered more efficiently. servational data. There is only a hint of a component on
For the radio pulse passing through the scattering re- the trailing edge of the pulse at 400 MHz (Hamilton et al.
gion theassumption of thestationary scattering seems reli- 1977), at 2.295 GHz it becomes more convincing (Downs
able. At the same time, the incident intensity and the pa- 1979; Krishnamohan & Downs 1983), and at 4.8 GHz
rameters of the scattering plasma are expected to fluctuate the trailing component is stronger than the precursor
from pulse to pulse. The resultant fluctuations of Γ affect (Gardner& Whiteoak 1969).
not only the observed radio profile shape but also the total Theradio observations ofLommen et al. (2007),which
intensity of the profile. Recall that the induced scattering haverevealedthecorrelationwiththehigh-energyemission,
leads to the intensity transfer between widely spaced fre- have been carried out at the frequency of 1 GHz. Unfortu-
quencies, ν′ = νθ2γ2/2 ν, and a substantial part of the nately,intheabovementionedpapertheradioprofileprop-
k ≫ ertieshavenotbeenpresentedindetailandtheauthorshave
lower-frequency intensity I(0) may come to the higher fre-
ν concentratedsolelyonthetimeofarrivaloftheradiopulses,
quency (cf. Eq. (13)). With the decreasing spectrum of the
assumingimplicitly thatthebasicfeaturesoftheradiopro-
pulsarradioemission,thismayimplyanoticeableincreaseof
filestructureanditsvariationswiththeradiopulseintensity
thetotalintensityofthehigher-frequencyprofileduetothe
arethesameasthosedescribed inKrishnamohan & Downs
strong precursor component resulting from the scattering.
(1983) for thedata at 2.3 GHz.
Thus, the induced scattering increases the radio profile in-
tensityatafixedfrequency,theamplificationbeingstronger
for higher Γ. 4.2 High-energy emission as a result of radio
All this is in line with the observed properties of the photon reprocessing
Vela’sradiopulses(Krishnamohan & Downs1983).Amore
Let us turn to the consequences of both the spontaneous
pronouncedprecursorcomponentisindeedcharacteristic of
scatteringoftheradiopulseandtheparticlesynchrotronre-
stronger pulses. At the same time, the rest of the profile
emissioninapplicationtotheVelapulsar.Asispointedout
weakens with the total intensity growth, being more effi-
above,ourconsiderationisrestrictedtothescatteringofthe
ciently suppressed by the induced scattering to higher fre-
A-polarization,sinceonlythismodeispresentintheVela’s
quencies.Instrongerpulses,wheretheroleofinducedscat-
radioemission.Thecharacteristicfrequencyofthescattered
teringis moresignificant, theregion of substantialsuppres-
radiation is given by ω′ η′ =ωη+s Ω s Ω, where
sion extendsto earlier pulse phases, where Γ becomes large max max ≈ max
η′=1/γ2,s =3y γ3/2istheharmonicnumbercorre-
enough. k max max 0
spondingtothespectralmaximumofthescatteredradiation
It should be noted that for any angle of incidence of and y =5/4. This is reduced to ω′ =1.5y Ωγ2γ and
max max 0
the radio beam the orientation of the scattered component
can beestimated as
is the same, θ′ 1/γ , and therefore the shape of the pre-
≈ k B γ γ r −3
cursor component should bedetermined bythevariation of ¯hω′=1.7 102y ⋆ 0 eV. (21)
× max1012G10310 108cm
the magnetic field orientation and the scattering efficiency
(cid:16) (cid:17)
across the scattering region. The magnetic field direction The characteristic frequency of the re-emitted radiation is
changes monotonically, so that the leading edge of the pre- given by Eq. (21) with ymax = 0.3. Given that B⋆ =
cursor is formed by theradiation at theleading edgeof the 3.4 1012 G we find ¯hωmax = 0.8 keV and 0.2 keV for
×
mainpulseandviceversa.Forafixedfrequencyofthescat- the scattered and re-emitted radiation, respectively. Thus,
tered radiation ν′ we have νθ2 =const, i.e. at different lon- thespectrum of thereprocessed radiation can extendup to
gitudes the precursor component is fed by the main pulse thesoft X-rayrange.
radiation of different frequencies, and the efficiency of the ProceedingfromEq.(11),whichgivesthescatteredand
component growth Γ ν−α θ2α. As the angle of inci- synchrotronpowersofasingleparticle,onecanestimatethe
∝ ∝
dence is larger for the leading rays (see Fig. 4), the peak of luminosities due to the scattering and re-emission, Lsc =
theprecursorisformedbytheradiationattheextremelead- (PAA+PAB)NeSr and Lsyn = (PsAyn +PsByn)NeSr. Using
ingedgeofthemainpulse,wheretheoriginalintensityiswell Eqs. (16), (18) and (19), we find
below the profile maximum but is large enough to provide
B 0.1s 5 L ν −4−α
an efficient scattering. Hence, in stronger pulses, where the L =2 1026 ⋆
sc × 1012G P 1029ergs−1 109Hz
scattering efficiency is higher, a significant contribution to
(cid:16) (cid:17) (cid:16) (cid:17)
thescatteredcomponentcomesfromthemainpulseraysat κ γ −4 γ0 6 r −20 ergs−1, (22)
theearlierpulselongitudesand,correspondingly,theprecur- × 102 103 10 108cm
sorarrivessomewhatearlier.Theobservationsdorevealthe (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
B P −1 ϑ 2
unambiguous relation between the arrival time of the pulse Lsyn =1029 1012⋆G 0.1s 0.1
and its total intensity (Krishnamohan & Downs 1983). In
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)
ourmodel, both quantities aredetermined bytheefficiency κ γ0 2 r ergs−1. (23)
of induced scattering, and the salient features of the radio × 102 10 108cm
profilestructureanditsfluctuationsareexplainednaturally. Recall (cid:16)that(cid:17)L(cid:16)is the to(cid:17)tal radio luminosity of the pulsar,
Thefrequencyevolutionofthepulseprofileisalsowor- α is the spectral index of the radio emission (α = 1.3 for
thy to be addressed. According to Eq. (20), the scatter- ν < 109 Hz and α = 2.7 for ν > 109 Hz). Taking into
ing efficiency is much larger at lower frequencies. There- account the parameters of the Vela pulsar, L = 2 1029
×
fore the precursor component should be most pronounced ergs−1,P =0.089s,B =3.4 1012 Gandϑ=0.15,yields
⋆
at lower frequencies, whereas the rest of the pulse should L 1029 ergs−1 andL ×1031 ergs−1.Thelattervalue
sc syn
∼ ∼
8 S. A. Petrova
isconsistentwiththeobservedluminosityoftheVelapulsar up to a few keV, where they turn into the trough. All this
in therange of 0.2–8 keV.(Pavlov et al. 2001). strongly supports an idea of their origin as a result of syn-
Our technique developed in Sect. 2 implies that the chrotronre-emissionandspontaneousscatteringbyspiraling
scattering is considered as a secondary process, and it is particles.
applicable as long as the total power scattered is less than Peak1ofthehigh-energyprofileisalsoofinterest,since
the synchrotron power, i.e. L /L 1. Comparison of its intensity is also affected by the radio pulse properties
sc syn
≪
Eqs. (22) and (23) shows that this inequality is generally (Lommen et al.2007).Itshouldbenotedthatintheoptical
valid, though both luminosities are strong functions of the – soft X-ray range peak 1 noticeably shifts with frequency
parameters. The role of the scattered emission is believed toward the radio pulse location, whereas at higher energies
to be most significant in the range of a few keV, where it keeps a fixed position (Hardinget al. 2002). The nature
the spectral power of the scattered radiation reaches max- of peak 1 seems questionable. Its position in the pulse pro-
imum, while the synchrotron spectrum drops exponentially file, 90◦aftertheradiopulse,excludesthemagnetospheric
∼
(see Fig. 1). originofthiscomponentprovidedthatthehigh-energyemis-
It is important to note that the luminosity due to sionisdirectedapproximatelyalongthemagneticfield.One
the scattering depends on the Lorentz-factor of the parti- can speculate, however, that the radiation forming peak 1
clegyrationmuchstrongerthanthesynchrotronluminosity: presents the synchrotron emission of the particles beyond
L γ6, whereas L γ2. The evolution of the particle the light cylinder. A more detailed analysis of the high-
sc ∝ 0 syn ∝ 0
Lorentz-factorγ isdeterminedbytheradiointensity,which energy profile of the Vela pulsar is beyond the framework
0
comes to the resonance region and causes the particle mo- of thepresent paper.
mentum evolution. Then it is the scattered luminosity L
sc
that is strongly sensitive to the radio emission characteris-
tics and theirfluctuations. Hence,thehigh-energy emission 4.3 Manifestations of the radio – high-energy
is believed to exhibit most pronounced correlation with ra- connection
dioemissionintherangebeyondthesynchrotronmaximum,
wherethecontributionofthescatteredpowerissubstantial. The physical connection of the high-energy radiation con-
This is indeed observed in theVela pulsar. sidered in Sect. 4.2 to the radio pulse pulse properties dis-
It should be noted that the particle distribution func- cussed in Sect. 4.1 is expected to have observational mani-
tion in momenta is sufficiently broad, and it is believed to festations. To analyse them let us first note that the spon-
evolve with distance significantly. Therefore the resultant taneous scattering into the high-energy range is most effi-
synchrotron spectrum do not resemble the single-particle cient at low enough altitudes, r rc (cf. Eq. (22)), due to
≪
spectrum:duetothestrongspatialdependenceofω′ the largernumberdensitiesofthescatteringparticles,Ne r−3,
spectral maximum is smeared, and the total spectmruaxm of and larger incident intensities, Iν r−2, in this r∝egion.
∝
the particles is modified substantially (for a more detailed The radio precursor component is formed at higher alti-
analysis see Petrova 2003). The same is expected for the tudes, r rc, where the rate of spontaneous scattering is
∼
scattered spectrum as well. Therefore in the present paper already less. Besides that, the precursor arises at a small
we do not analyze the spectral slopes. angle θ′ ∼ 1/γk to the ambient magnetic field, and, corre-
spondingly, the scattering cross-section for the incident A-
Now let us consider the location of the scattered and
polarizationisless(cf.Eq.(11)).Inaddition,theorientation
re-emitted radiation in the pulse profile. Both components
of the precursor with respect to the ambient magnetic field
of the high-energy emission resulting from the radio pho-
rapidly changes to θ′ = r/2r 1/γ , so that the origi-
ton reprocessing are believed to arise at an angle ∼ 1/γk nal under-resonance radiationL, ω≫′/γ2 k Ω, passes through
to the ambient magnetic field. Then the high-energy emis- k ≪
sion generated close enough to the radio emission region the resonance, ω′θ′2/2 = Ω, very soon. Therefore it is the
should appear in the profile at the original radio pulse po- mainpulsethatchieflycontributestothespontaneousscat-
sition. The high-energy emission originating at higher alti- tering to high energies. In contrast to the resonant absorp-
tudesshouldappear r/2r aheadofthemainpulseinra- tion, this process does not affect the radio pulse intensity
L
∼
dio (seeSect. 4.1). In particular, theradiation generated at considerably (Petrova 2008a). Note also that the resonant
thealtitudesof theorderof thecyclotron resonance radius, absorption suppresses the main pulse and precursor alike,
r r , should coincide in phase with the radio precursor so that thesynchrotron re-emission is related to both radio
c
∼
position.Theemissionregionofthehigh-energycomponent components equally.
whichresultsfromthescatteringoftheunder-resonancera- Thus, the main pulse of the radio profile is subject to
dio photons is restricted to r r , and, correspondingly, boththespontaneousandinducedscatterings.Thefirstpro-
c
∼
this component is present only in the same phase range as cess contributes to the soft X-ray component roughly co-
the radio precursor. This range is usually classified as the incident with the radio pulse window, whereas the second
position of peak 4in thehigh-energy profile(Harding et al. process gives rise to the precursor component on the radio
2002). As for the synchrotron emission, it can be efficient profile. The efficiencies of the two processes are expected
over a wider range of altitudes, even beyond the resonance to vary from pulse to pulse because of fluctuations of the
region,anditisexpectedtocontributesignificantlytoboth plasma parameters. This is believed to underlie the obser-
peak4andpeak3,whichprecedestheradiopulseby 30◦. vational manifestation of the X-ray – radio connection in
∼
Notethattheposition ofpeak3impliesthecomponentori- theVelapulsar. Weakerinducedscattering implies less effi-
gin close to the light cylinder. It is important to point out cientintensitytransferfromthemainpulsetotheprecursor.
that,incontrasttotherestofthehigh-energyprofile,peaks Then the resultant intensity of the radio pulse is less (see
3and4arepresentonlyintheopticalandsoftX-rayrange, Sect.4.1),theprecursorisweakerandthemainpulseisless
X-ray – radio correlation in the Vela pulsar 9
suppressed.Inthiscase,themainpulsecanmoreefficiently The radio pulse is also subject to spontaneous scatter-
participateinthespontaneousscattering, andtheresultant ing by the spiraling particles. The photons below the res-
high-energycomponentshouldbemorepronounced.Thisis onance, ωη Ω, are chiefly scattered to high harmonics
≪
in line with the observed trend: in the Vela pulsar weaker of the particle gyrofrequency, s γ3. We have analyzed
∼ 0
radiopulsesareaccompaniedbystrongerhigh-energyemis- the characteristics of the scattered radiation in detail and
sion at theposition ofpeaks3and4(Lommen et al. 2007). compared them with those of the synchrotron radiation of
Onecanspeculatethatthefluctuationsoftheeffectivevalue the same particle. In particular, it is found that the scat-
of γ make the dominant contribution to the variations of tered power peaks at somewhat higher energies than the
0
the efficiencies of the spontaneous and induced scatterings. synchrotronone.Inapplication totheVelapulsar, theesti-
At larger γ the induced scattering is weaker (see Eq. (20)) matesofthespectralmaximayield 0.8keVand 0.2keV,
0
∼ ∼
andthehigh-energyluminositiesL andL arelarger(see respectively. The synchrotron luminosity is 1031 ergs−1
sc syn
∼
Eqs. (22)-(23)). and well agrees with the observed values in the soft X-ray
Less values of γ mean weaker momentum evolution of band(e.gPavlov et al.2001).Althoughthetotalluminosity
0
the particles in the course of resonant absorption of radio provided by the scattering is less, in the range beyond the
emission. Then a greater part of the particle gyration en- synchrotron maximum the scattered power may still con-
ergyisexpectedtobere-emittedathigheraltitudes,beyond tribute substantially. It is important to note an extremely
thelightcylinder,contributingtopeak1ofthehigh-energy strong dependence of the scattered power on the Lorentz-
profile. Hence, strong radio pulses with more pronounced factor of the particle gyration, L γ6, which means a
sc ∝ 0
precursors, which correspond to large Γ, should be accom- strong relation of the scattered component to the radio in-
panied by stronger high-energy emission at the position of tensity.
peak 1 and weaker emission in the trough. This also agrees Similarlytothesynchrotronemission,thescatteredra-
with theobserved trend. diation concentrates close to the ambient magnetic field di-
rection,θ′ 1/γ .Giventhatthesynchrotronandscattered
∼ k
emissionsoriginatewellabovetheradioemissionregionand
insidethelightcylinder,theyshouldprecedethemainradio
5 SUMMARY AND DISCUSSION
pulse by . 30◦. This position can be identified with that
We have considered the processes of spontaneous and in- of peaks 3 and 4 of the soft X-ray profile, which turn into
ducedscattering off theparticles performing relativistic he- thetrough at somewhat higherenergies (e.g. Harding et al.
licalmotioninanexternalmagneticfield.Thetheoryisap- 2002). If the synchrotron re-emission continues beyond the
plied to the radio wave scattering off the secondary plasma light cylinder, it may contributeto peak 1 of theprofile.
particlesinthepulsarmagnetosphere.Theparticlesarebe- TheradioemissionoftheVelapulsarisbelievedtopar-
lieved tohavesubstantialgyration energies duetoresonant ticipate in both the spontaneous and induced scatterings,
absorptionoftheradioemissionintheoutermagnetosphere. and the interplay between these processes can account for
InapplicationtotheVelapulsar,theinducedscattering the observed X-ray – radio correlation. If the main pulse is
ofradio wavesbetween thestateswell below theresonance, less suppressed by the induced scattering to the precursor,
ωη = ω′η′ Ω, can be efficient. An extremely bright and it is more efficiently scattered to high energies. This is con-
≪
narrowradiobeamisscatteredchieflyintothebackground, sistent with the observations: weaker radio pulses with less
in the direction corresponding to the maximum scattering pronouncedprecursors areaccompanied bythehigh-energy
probability. The scattered component is directed approxi- pulseswith strongeremission in thetrough (Lommen et al.
mately along the ambient magnetic field, θ′ 1/γ , and 2007).Giventhatthevariationsofthescatteringefficiencies
max ∼ k
appearsintheradioprofileasaprecursortothemainpulse. are determined by the fluctuations of γ0, larger γ0 imply
This scenario for the first time allows to explain the main lessefficientinducedscatteringandsimultaneouslystronger
featuresoftheradioprofilestructureoftheVelapulsarand high-energyluminosities.Forsmaller γ0 thesynchrotronre-
its pulse-to-pulsefluctuations. emission is expected to continue beyond the light cylinder,
In the case considered, the induced scattering trans- contributing to peak 1 of the high-energy profile. At the
fers the main pulse intensity to the higher frequencies, same time, smaller γ0 imply more efficient induced scatter-
ω′ ωθ2γ2 ω. With the decreasing spectrum of the ing, a more pronounced radio precursor and stronger resul-
∼ k ≫
pulsarradioemission, thisimpliesintensityenhancementof tantradiopulses.Allthisisinlinewiththeobservedtrends
the radio profile at a fixed frequency. Thus, larger scatter- (Krishnamohan & Downs1983;Lommen et al.2007).Thus,
ingefficiencies should result inboth higherpulseintensities our model explains the salient features of the radio profile
and more pronounced precursors. Such a correlation is re- formation of the Vela pulsar, the peculiarities of its soft X-
ally observed in the Vela pulsar (Krishnamohan & Downs rayprofileaswellastheobservedX-ray–radioconnection.
1983). Moreover, stronger scattering means that a larger It should be noted that our model of the radio profile
partofthemainpulsecanbescatteredefficiently,theregion formation in the Vela pulsar is too simplified, since it in-
of substantial suppression extending from the very trailing cludes only two components, the main pulse and precursor.
edge of the profile toward earlier phases. As the scattering Krishnamohan & Downs (1983) have established four com-
of the leading edge of the main pulse makes the dominant ponentsof theVela’s radio profile. Later on Johnston et al.
contribution to the precursor formation, higher scattering (2001) and Kramer et al. (2002) have discovered sporadic
efficiencies imply not only larger intensities but also earlier activityat theleadingedgeofthepulse(theso-called giant
phases of the precursor peak. The observations do reveal micropulses)andinthebumpregion inthetrailingpartof
early arrival of the precursor component in strong pulses the profile. Further development of our model is needed in
(Krishnamohan & Downs 1983). ordertoincludethesephenomena.However,webelievethat
10 S. A. Petrova
thesepeculiaritiesdonotaffecttheX-ray–radioconnection REFERENCES
noticeably.
Blandford R. D., Scharlemann E. T. 1976, MNRAS, 174,
We have considered the induced scattering in the ap- 59
proximation of a strong magnetic field. This process can Canuto V., Lodenquai J., Ruderman M. 1971, Phys. Rev.
be efficient only if the incident and scattered waves have D,3, 2303
the ordinary polarization (the A-polarization), i.e. if their Downs G. S. 1979, ApJS, 40, 365
electric vectors are in the plane of the ambient magnetic Gardner F. F., Whiteoak J. B. 1969, Nature, 224, 891
field. In the Vela pulsar, only one polarization mode is GilJ.A.,KhechinashviliD.G.,MelikidzeG.I.2001,ApJ,
present. Various emission theories used to identify it with 551, 867
theordinarymode.Thepointisthattheextraordinary(B-) Gouiffes C. 1998, in Neutron Stars and Pulsars, ed. N.
mode has the vacuum dispersion, and its direct generation Shibazaki, N. Kawai, S. Shibata, T.Kifune (Tokyo: Univ.
by any plasma mechanism seems problematic. The recent Acad.Press), 363
high-energy observations of the Vela’s pulsar wind nebula HamiltonP.A.,McCullochP.M.,AblesJ.G.,Komesaroff
have cast some doubt as to the type of the radio polar- M. M. 1977, MNRAS,180, 1
ization in this pulsar (Lai et al. 2001; Helfand et al. 2001; Harding A. K., Strickman M. S., Gwinn C., et al. 2002,
Radhakrishnan & Deshpande2001).Theyhaverevealedthe ApJ,576, 376
jet in the direction of the pulsar proper motion. If this jet Harding A. K., Usov V. V., Muslimov A. G. 2005, ApJ,
is directed along the pulsar rotational axis, then the radio 622, 531
emissionhasextraordinarypolarization.Notethatjetalign- Harding A. K. 2008, in Astrophysics of Compact Objects:
ment with therotational axis is well ascertained for theac- InternationalConferenceonAstrophysicsofCompactOb-
cretion systems, but the radio pulsars do not seem to be jects. AIP Conference Proceedings, 968, 104
such systems, and the nature of the Vela’s jet is obscure. Harding A. K., Stern J. V., Dyks J., Frackowiak M. 2008,
Thereforewehavestillassumedtheordinarypolarizationof eprint arXiv:0803.0699
theVela’s radio emission. Helfand D. J., Gotthelf E. V., Halpern J. P. 2001, ApJ,
556, 380
Thehigh-energyemissionoftheVelapulsaristoocom-
Johnston S., van Straten W., Kramer M., Bailes M. 2001,
plicated, and the emission mechanisms involved cannot be
ApJ,549, L101
exhaustedbythoseconsideredinthepresentpaper.Asisar-
Kanbach G., Arzoumanian Z., Bertsch D. L., et al. 1994,
gued by Harding et al. (2002), the Vela’s high-energy emis-
A&A,289, 855
sion consists of the soft and hard components. The radio
photon reprocessing contributes solely to the soft compo- Kramer M., Johnston S., van Straten W. 2002, MNRAS,
nent. The photons emitted inside the light cylinder appear 334, 523
in the phase region preceding the radio pulse by . 30◦. Krishnamohan S., Downs G. S. 1983, ApJ, 265, 372
The synchrotron re-emission beyond the light cylinder may Lai D.,Chernoff D.F., Cordes J. M.2001, ApJ,549, 1111
at least partially contribute to peak 1, whereas peak 2 is Lommen A.,Donovan J., Gwinn C., et al. 2007, ApJ,657,
thought to result from some othermechanism. 436
Lyubarskii Yu. E., Petrova S. A., 1996, Astron. Let., 22,
Within the framework of our model, the peculiar X-
399
ray – radio correlation observed in the Vela pulsar is only
LyubarskiiYu.E., PetrovaS. A. 1998, A&A,337, 433
characteristic of a narrow spectral range in the soft X-ray
Melrose D. 2004, in Young Neutron Stars and Their En-
band,namelyjustbeyondthespectralmaximumofthehigh-
vironments, IAU Symposium no. 218, held as part of
energy component roughly coincident with the radio pulse
the IAU General Assembly, 14-17 July, 2003 in Sydney,
window. It is the range where the contribution of the scat-
Australia. Edited by Fernando Camilo and Bryan M.
tering off the spiraling particles may be substantial, and
Gaensler.SanFrancisco,CA:AstronomicalSocietyofthe
the scattered component is tightly connected to the radio
Pacific, p.349
pulseintensity.Atthesametime,anotherprocessofthera-
PavlovG.G.,ZavlinV.E.,SanwalD.,BurwitzV.,Garmire
dio photon reprocessing to high energies – the synchrotron
G. P.2001, ApJ, 552, L129
re-emission of the spiraling particles – is believed to be sig-
Petrova S.A. 2002, MNRAS,336, 774
nificant over a wider spectral range. Further observational
Petrova S.A. 2003, MNRAS,340, 1229
studies of the X-ray – radio correlation at softer energies
Petrova S.A. 2008a, MNRAS,383, 1413
would be of interest. A more detailed comparison of the
Petrova S.A. 2008b, MNRAS,384, L1
high-energy data with the radio profile properties would be
Petrova S.A. 2008c, MNRAS,385, 2143
particularly useful.
RadhakrishnanV.,DeshpandeA.A.2001, A&A,379, 551
Theradiophotonreprocessing isbelievedtotakeplace Romani R. W., Kargaltsev O., Pavlov G. G. 2005, ApJ,
in other pulsars as well. The resultant radio – high-energy 627, 383
connectionisyettobediscovered.Itshouldbekeptinmind Sanwal,D.,PavlovG.G.,KargaltsevO.Y.,etal.2002,in
thattheobservationalmanifestationsofthisconnectionmay ASP Conf. Proc. 271, Neutron Stars in Supernova Rem-
be quite different because of the difference of the physical nants, ed. P. O. Slane & B.M. Gaensler, (San Francisco:
conditions and parameters. In the radio range, pulsars are ASP),353
known to exhibit various fluctuation phenomena, and the ShearerA.,StappersB.,O’ConnorP.,etal.2003, Science,
simultaneousobservationsintheradioandsoft high-energy 301, 493
ranges are expected to reveal diversiform connections. Shibanov Yu. A., Koptsevich A. B., Sollerman J.,