X-ray pulsar radiation from polar cap heated by back-flow bombardment

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by  J. Gil

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
X-ray pulsar radiation from polar cap heated by back-flow
bombardment
J. Gil,1 G. Melikidze,1,2 B. Zhang3
1Institute of Astronomy, University of Zielona G´ora, Lubuska 2, 65-265, Zielona G´ora, Poland
7 2Abastumani Astrophysical Observatory, Al. Kazbegi ave. 2a, 0160, Tbilisi, Georgia
0 3Department of Physics, University of Nevada, Las Vegas, USA
0
2

n
a
J
ABSTRACT

6
We consider the problem of the thermal X-ray radiation from the hot polar cap of
1 radio pulsars showing evidence of E×B subpulse drift in radio band. In our recent

Paper I, using the partially screened gap (PSG) model of inner acceleration region
2
we derived a simple relationship between the drift rate of subpulses observed in a
v
radio-bandandthethermalX-rayluminosityfrompolarcapsheatedbytheback-flow
6
2 particle bombardment. This relationship can be tested for pulsars in which the so-
2 called carousel rotation time P4, reflecting the E×B plasma drift, and the thermal
2 X-ray luminosity Lx from the hot polar cap are known. To test the model we used
1 only two available pulsars: PSRs B0943+10 and B1133+16. They both satisfied the
6 modelprediction,althoughdue to lowphotonstatistics the thermalcomponentcould
0 not be firmly identified from the X-ray data. Nevertheless,these pulsars were at least
/
h consistent with PSG pulsar model.
p Inthepresentpaperweconsidertwomorepulsars:PSRsB0656+14andB0628-28,
- whose data have recently become available. In PSR B0656+14 the thermal radiation
o
from the hot polar cap was clearly detected, and PSR B0628-28 also seems to have
r
t such a component.
s
a Inallcasesfor whichbothP4 andLx arepresentlyknown,the PSGpulsarmodel
: seems to be fully confirmed. Other available models of inner acceleration region fail
v
to explain the observedrelationshipbetween radio and X-raydata. The pure vacuum
i
X gapmodel predicts too high Lx and too lowP4, while the space charge limited model
r predicts too low Lx and the origin of the subpulse drift has no natural explanation.
a
Key words: pulsars:pulsars: individual: B0628-28;B0656+14;0943+10;B1133+16
– X-rays: thermal
1 INTRODUCTION drifting mechanism to work cannot be very different from
theemission mechanism of radio pulsars.
Althoughalmost40yearshavepassedsincethediscoveryof It is therefore likely that the drifting subpulse phe-
pulsars, the mechanism of their coherent radio emission is nomenonoriginatesfromtheso-calledinneraccelerationre-
stillnotknown.ThetheoryofpulsatingX-rayemissionalso gion right abovethepolar cap, which powers thepulsar ra-
demands further development. The puzzling phenomenon diation. In the classical model of Ruderman & Sutherland
of drifting subpulses is widely regarded as a powerful tool (1975; RS75 henceforth) the subpulse-associated spark fila-
fortheinvestigationofthepulsarradiationmechanism.Re- ments of plasma circulate in the purea Vacuum Gap (VG)
cently, this phenomenon received a lot of attention, mostly around the magnetic axis due to the E×B plasma drift.
owing tothenewly developed techniquesfor theanalysis of This model is widely regarded as a natural and plausible
thepulsarradioemission fluctuations(Edwards&Stappers explanation of the drifting subpulse phenomenon, at least
2002, 2003). Using these techniques, Weltevrede, Edwards, qualitatively. On the quantitativelevel, this model predicts
& Stappers (2006a, WES06 henceforth) presented the re- too high a drifting rate, or too short a period P4 (Pˆ3 in the
sultsofthesystematic,unbiasedsearchforthedriftingsub- nomenclature introduced by RS75), of the sparks’ circula-
pulsesand/orphasestationaryintensitymodulationsinsin- tion around the polar cap, as compared with the observa-
gle pulsesof alarge sample ofpulsars. They found that the tions (e.g. Deshpande & Rankin, 1999; DR99 henceforth).
fraction of pulsars showing evidence of drifting subpulses Also, the predicted heating rate of the polar cap surface
is at least 55 % and concluded that the conditions for the due to the spark-associated back-flow bombardment is too
2 J. Gil, G.I. Melikidze, & B. Zhang
high. The alternative model, namely the space charge lim- ThePSGmodelcanbetestediftwoobservationalquan-
ited model (SCLF; e.g. Arons & Sharleman 1979), predicts tities are known: (i) the circulational period P4 for drifting
too low a heating rate and has no natural explanation for subpulsesobservedintheradioband(alsocalled thepulsar
the phenomenon of drifting subpulses (Zhang & Harding carousel time), and (ii) the X-ray luminosity L of thermal
x
2000; Harding & Muslimov 2002). However, this model has black-body(BB) radiation from thehotpolarcap(seeEqs.
an advantage over the VG model, namely it is free of the 2and3below).Theabovementionedobservationsof PSRs
so-called binding energy problem, to avoid which the VG B0943+10 and B1133+16 are not decisive. Indeed, due to
model requires an ad hoc assumption of the strong, non- poor photon statistics, their spectra can be described by
dipolarsurfacemagneticfield(forreviewandmoredetailed eitherathermalmodel,anon-thermalmodel,oracombina-
discussion see Gil & Melikidze 2002). tion of theboth.In anycase, onecan posetheupperlimits
Motivated by these observational discrepancies of the forthethermalradiation from thehot polarcapfrom these
otherwise attractive VG model, Gil, Melikidze & Geppert data,sothat thePSGmodelcould betestedat least in the
(2003;GMG03henceforth)developedfurthertheideaofthe order of magnitude approximation.
inner acceleration region above the polar cap by including Inthispaperweincludetwomorepulsarsforwhichval-
thepartialscreeningcausedbythethermionicionsflowfrom uesofbothP4andLxarecurrentlyknown:PSRsB0656+14
thesurface heated by sparks. We call this kindof the inner andB0628-28. Theformer casewas arealbreakthroughfor
accelerationregionthe”partiallyscreenedgap”(PSGhence- our considerations and testing. Indeed, while in the other
forth) 1 Since the PSG potential drop is much lower than cases thecharacter of the spectrum was not certain, in this
that in the RS75 model, the intrinsic drift rate P4 is com- pulsar(one of theThree Musketeers) thethermal radiation
patible with the observations. This is a consequence of the from the hot polar cap was clearly detected (De Luca et
reducedpotentialdrop,partiallyscreenedbythethermionic al. 2005). PSR B0628-28 was observed with Chandra and
ion flow from the polar cap surface. In the pure vacuum XMM-Newton observatories byTepedelenlioˇglu & O¨gelman
RS75 gap, the heating of the polar cap is definitely too in- (2005; hereafter TO¨05). Weshow that both pulsars comply
tense (e.g. Zhang, Harding & Muslimov 2000; Zhang, San- thePSG model, increasing the numberof pulsars that pass
wal & Pavlov 2005; ZSP05 henceforth).On theotherhand, themodeltestexpressedbyEqs.(2)and(3)fromtwotofour.
the SCLF model predicts too low a heating rate as com- Atthemoment,PSRsB0943+10,B1133+16,B0656+14and
pared with observations (Zhang & Harding 2000; Harding B0628-28aretheonlypulsarsforwhichbothP4 andLx are
& Muslimov 2002). Thus, by measuring the thermal X-ray known. It is important to show that all of them follow the
luminosity from heated polar caps one can potentially re- theoretical prediction curvein Fig. 1.
veal the nature of the inner acceleration region in pulsars.
This can also help to understand a mechanism of drifting
subpulses, which appears to be a common phenomenon in
2 PSG MODEL OF THE INNER
radio pulsars.
ACCELERATION REGION
ZSP05 were the first who attempted to test different
available models of theinner acceleration region in pulsars, Thechargedepletedinneracceleration regionabovethepo-
using a concept of the polar cap heated by the back-flow lar cap results from the deviation of a local charge density
particlebombardment.Theyobservedthebeststudieddrift- ρfrom theco-rotational chargedensity(Goldreich &Julian
ing subpulse radio pulsar PSR B0943+10 with the XMM- 1969) ρGJ = −Ω·Bs/2πc ≈ Bs/cP. For isolated neutron
Newton observatory and argued that the detected X-ray stars onemight expectthesurface toconsist mainly of iron
photons were consistent with PSG formed in the strong, formedattheneutronstar’sbirth(e.g.Lai2001).Therefore,
non-dipolar magnetic field just above the surface of a very the charge depletion above the polar cap can result from
small and hot polar cap. Recently Gil, Melikidze & Zhang binding of the positive 56Fe ions (at least partially) in the
26
(2006 a,b; hereafter Paper I and II, respectively) developed neutronstarsurface.Ifthisisreallypossible(seeMendin&
a detailed model for thethermal X-rayemission from radio Lai2006,andPaperIIfordetails),thenthepositivecharges
driftingpulsars.TheyappliedtheirmodeltoPSRB0943+10 cannot be supplied at the rate that would compensate the
aswell as toPSRB1133+16, which was observedin X-rays inertialoutflowthroughthelightcylinder.Asaresult,asig-
withChandraobservatorybyKargaltsev,Pavlov&Garmire nificant part of the unipolar potential drop develops above
(2006, KPG06 henceforth). These authors found that this thepolarcap,whichcanacceleratechargedparticlestorela-
case is also consistent with the thermal radiation from a tivisticenergiesandpowerthepulsarradiation mechanism.
small hot spot, much smaller than the canonical polar cap. The ignition of cascading production of the electron-
PSRB1133+16isalmostatwinofPSRB0943+10interms positron plasma is crucial for limitation of the growing po-
ofP andP˙ valuesand,interestingly,bothpulsarshavevery tential drop across the gap. The accelerated positrons will
similar X-raysignatures, inagreement withthePSGmodel leave the acceleration region, while the electrons bombard
(see Table 1 and Fig. 1). the polar cap surface, causing a thermal ejection of ions.
This thermal ejection will cause partial screening of theac-
celeration potential drop ∆V corresponding to a shielding
1 Cheng&Ruderman(1980) werethefirsttoconsider thePSG tfahcetochraηrg=ed1e−nsρitiy/ρoGfJth(eseeejeGctMedGi0o3nsfo,r∆dVet=ailηs)(,2πw/hcePre)Bρihi2s
model.However,theyarguedthatevenwithpartialscreeningin- s
is thepotential drop and h is the height of theacceleration
cluded,theconditionsabovethepolarcapareclosetopureVGas
inRS75.GMG03demonstratedthattheactualthermostaticself- region.Thegap potentialdropiscompletely screenedwhen
regulationestablishestheacceleratingpotentialdropthatmaybe thetotalchargedensityρ=ρi+ρ+reachestheco-rotational
aslowasfewpercentofthatofRS75value. valueρGJ.
X-ray pulsar radiation from polar cap heated by back-flow bombardment 3
GMG03 argued that the actual potential drop ∆V misleadingvaluesofaforsomeuntypicalpulsars(seediscus-
shouldbethermostaticallyregulatedandthereshouldbees- sion insection 4).However,usingthisconceptwecanwrite
tablishedaquasi-equilibriumstate,inwhichheatingdueto the shielding factor in the form η ≈ (1/2π)(P/P3), which
electron bombardment is balanced by cooling due to ther- depends only on a relatively easy-to-measure primary drift
mal radiation. The quasi-equilibrium condition is Qcool = periodicity P3.Notealso thatP4/P =a/(2η). Weshow the
Q , where Q = σT4 is the cooling power surface values of the model parameters obtained from these equa-
heat cool s
density by thermal radiation from the polar cap surface tions in Table 1.
and Q = γm c3n is the heating power surface den- The X-ray thermal luminosity from the polar cap
heat e
sity due to back-flow bombardment, γ = e∆V/m c2 is the with a temperature T is L = σT4πr2 = 1.2 ×
e s x s p
Lorentz factor, n = nGJ −ni = ηnGJ is the number den- 1032(P˙−15/P3)(ηh/rp)2 erg/s, which can be compared
sity of the back-flowing particles that deposit their kinetic with the spin-down power E˙ = IΩΩ˙ = 3.95I45 ×
energy at the polar cap surface, η is the shielding factor, 1031P˙−15/P3 erg/s, where I =I451045g cm2 is the neutron
ni is the charge number density of the thermionic ions and starmomentofinertia2andI45 =1+−10..2252.Usingequation(1)
nGJ = ρGJ/e = 1.4×1011bP˙−0.155P−0.5cm−3 is the corota- we can derivetheformula for thermal X-rayluminosity as
tional charge number density. It is straightforward to ob-
tainanexpressionforthequasi-equilibriumsurfacetemper- Lx =2.5×1031(P˙−15/P3)(P4/P)−2, (2)
atureintheformTs =(6.2×104K)(P˙−15/P)1/4η1/2b1/2h1/2, or in the simpler form representing the efficiency with re-
where the parameter b = Bs/Bd = Apc/Abol describes the spect tothe spin-down power
domination of the local actual surface magnetic field over
thecanonicaldipolarcomponentatthepolarcap,andP˙−15 Lx = 0.63 P4 −2. (3)
is the normalized period derivative. Here Apc = πrp2c and E˙ (cid:16) I45 (cid:17)(cid:16)P (cid:17)
Abol=Ap =πrp2 istheactual(bolometric) emittingsurface This equation is very useful for a direct comparison with
area, with rpc and rp being the canonical (RS75) and the the observations, since it contains only the observed quan-
actualpolarcapradius,respectively.Sincethetypicalpolar tities (although it is subject to a small uncertainty factor
cap temperature is Ts ∼106 K (Paper II), the actual value related to the unknown moment of inertia2), and it does
of b must be much larger than unity, as expected for the not dependon anydetails of thesparking gap model. Itre-
highly non-dipolar surface magnetic fields. flects the fact that both the subpulse drifting rate (due to
Theacceleratingpotentialdrop∆V andtheperpendic- E×Bplasma drift) andthepolar cap heatingrate (dueto
ular (with respect of the magnetic field lines) electric field back-flowbombardment)aredeterminedbythesamephys-
∆E,whichcausesE×Bdrift,mustberelatedtoeachother, ical quantity, which is the potential drop across the inner
and this relationship should be reflected in combined radio acceleration region just abovethe polar cap.
and X-ray data of pulsars showing drifting subpulses. This The microscopic properties of PSG model require a
is basically a conal phenomenon (Rankin 1986), so we can more sophisticated analysis, like the one presented in our
restrict ourselves to the periphery of the polar cap, where Paper II. Here we can give simplified but more intu-
these two potential drops are numerically equal to each itive estimate of the screening factor η = (a/2)(P/P4) =
other. Moreover, following the original “pillbox” method of (1/2π)/(P/P3) and the number of sparks N = (P4/P3) =
RS75 we can argue that the tangent electric field is strong πa, using arguments based on the complexity parameter
only at the polar cap boundary where ∆E = 0.5∆V/h = a=r /h presented in theparagraph below equation (1).
p
η(π/cP)B h (see Appendix A in GMG03 for details). Due
s
totheE×Bdriftthedischargeplasmaperformsaslowcir-
cumferential motion with velocity vd =c∆E/Bs =ηπh/P. 3 OBSERVATIONAL VERIFICATION
The time interval to make one full revolution around the
polar cap boundary is P4 ≈2πrp/vd. Onethen has Table 1 presents the observational data and the predicted
values of a number of quantities for four pulsars, which we
P4 = rp . (1) believe to show clear evidence of thermal X-ray emission
P 2ηh
fromthespark-heatedpolarcapsaswellastheyhaveknown
If the plasma above the polar cap is fragmented into fil- valuesofthecirculationalsubpulsedriftingperiodicity.The
aments (sparks), which determine the intensity structure predicted values of P4 and/or Lx are computed from equa-
of the instantaneous pulsar radio beam, then in principle, tion(3).ErrorsinLx istakenfromtheobservationalpapers
the circulational periodicity P4 can be measured/estimated orderivedfromthedistanceuncertainty(takenfromCordes
from the pattern of the observed drifting subpulses (Desh- &Lazio(2002),exceptthecaseofB0656+14forwhichitwas
pande & Rankin 1999, Gil & Sendyk 2003). According to obtainedbyBriskenetal.(2003)usingthepulsarparallax),
RS75, P4 = NP3, where N is the number of sparks con- whichever is greater. The relationship expressed by equa-
tributingtothedriftingsubpulsepatternobservedinagiven tion(3)isrepresentedbythesolidcurveinFig.1,withtwo
pulsar and P3 is the primary drift periodicity (distance be-
tweentheobservedsubpulsedriftbands).Ontheotherhand
N ≈2πr /2h=πa,wherethecomplexityparametercanbe 2 Consideringgeneralrelativity, themomentofinertiaofaneu-
estimatedp from the approximate formula a = 5P˙0.29P−0.64 tron star can be written as I = 0.21MR2/(1−2GM/(c2R)),
−15 whereM andRistheneutronstarmassandradius,respectively
(Gil & Sendyk, 2000; GS00 henceforth). One has to real-
(Ravenhall & Pethick, 1994). Taking M = 1.4 solar masses and
izethatthisapproximation wasderivedunderaspecificas- Rrangingfrom8×105to1.7×106cm,forthesoftestandstiffest
sumption concerning the actual surface magnetic field (see equationsofstate,respectively,oneobtainsthemomentofinertia
discussion below equation (11) in GS00), and it can give rangingfrom7.82×1044 to2.25×1045gcm2,respectively.
4 J. Gil, G.I. Melikidze, & B. Zhang
Table 1.Theobservedandthemodelparametersforthefourpulsars.
Name a P4/P P3/P η N Lx×10−28 erg s−1
(cid:0) (cid:1)
PSRB Obs. Pred. P/2πP3 aP/2P4 ⌊P4/P3⌋ ⌊πa⌋ Obs. Pred.
0628−28† 7.61 7+1 6+1 0.29+0.04 0.54+0.09 24+8 23 287+152 189+100
−1 −1 −0.04 −0.07 −6 −82 −54
0656+14† 29.1 20+1 18+2 0.22+0.01 0.73+0.04 90+10 91 5700+652 6037+652
−1 −2 −0.01 −0.03 −8 −561 −561
0943+10 6.73 37.4+0.4 36+8 1.87 0.085 0.09 20+1 21 5.1+0.6 4.7+2.0
−1.4 −6 −1 −1.7 −1.3
1133+16 6.52 33+3 27+5 3+2 0.05+0.11 0.10+0.01 11+25 20 6.8+1.1 5.3+1.1
−3 −2 −2 −0.02 −0.01 −5 −1.3 −0.8
Note:† AsP3 wasnotmeasuredforthesetwopulsarsweusedtheestimateof η tocalculateP3/P.
dashed curves describing the uncertainty in determining of
theneutronstarmomentofinertia2.TosavespaceinTable B 0 628-28
1wegivethebasicpulsarparameters(P,P˙−15,E˙×10−32,D)
next to the pulsar name in the paragraphs describing each 10
30
case below. 1

PSR B0943+10 (P = 1.099 s, P˙−15 = 3.49, E˙ = ·.L/E ) x B0656+ 1 4
1.04×1032 erg/s, D = 0.631+0.113 kps) is the best stud- (
−0.104 B1133+16
ied drifting subpulse radio pulsar. As this case, along with 1
PSRB1133+16,wasdiscussedearlierinPapersIandII,we
do not find it necessary to review it again (see Table 1 and
Fig. 1). Error bars for P4 were given by Rankin & Suley- B0943+10
manova (2006), while errors for L were given by ZSP05.
x 20 40
See section 4 for discussion. P / P
4
PSR B1133+16 (P = 1.188 s, P˙−15 = 3.73, E˙ = Figure1.TheefficiencyofthermalX-rayemissionfromhotpo-
0.88×1032 erg/s, D = 0.35+−00..0022 kps) is almost a twin of larcapLxversuscirculationperiodP4ofdriftingsubpulsesinthe
PSR B0943+10, in both radio and X-ray bands as it was radioband.ThesolidcurverepresentsthepredictionofthePSG
demonstrated in Papers I and II,(see Table 1 and Fig. 1). model (equation 3) with I45 =1, whilethe dotted curves corre-
ErrorbarsforP4 weregivenbyWES06,whileerrorsfor Lx spondtouncertainties indeterminingofthemomentofinertia1.
were given byKPG06. See section 4 for discussion. ErrorbarsonP4 andLx weregivenbytheauthorsmentionedin
thetext.
PSR B0656+14 (P = 0.385 s, P˙−15 = 55.0, E˙ =
381 ×1032 erg/s, D = 0.288+0.033 kps) is one of the fa-
−0.027
mous Three Musketeers, in which the thermal X-ray emis-
sionfromthehotpolarcapwasclearlydetected(DeLucaet
al.2005).Thispulsarisverybright,sothephotonstatistics
aregoodenoughtoallowidentificationoftheBBcomponent
in the spectrum. As indicated in Table 1, the X-ray lumi-
nosity of this hot-spot BB component is Lx ∼ 5.7×1031 PSR B0628−28 (P = 1.244 s, P˙−15 = 7.12, E˙ =
ergs/s. This value, when inserted into equation (2), returns 1.46×1032 erg/s, D = 1.444+−00..226757 kps) is an exceptional
the predicted value of P4 = 20.6P. Amazingly, Weltevrede pulsaraccording to TO¨05. ItsX-rayluminosity exceeds the
etal.(2006b) reported recentlytheperiodicity of(20±1)P maximumefficiencylinederivedbyPossentietal.(2002)by
associated with thequasi-periodicamplitudemodulation of alargefactor.However,oneshouldnotethatPSRB0943+10
erratic and strong emission from this pulsar. Thus, it is with its luminosity derived from the PL fit, also exceeds
tempting to interpreted this period as the circulation time thismaximum efficiency(ZSP05,TO¨05). TheBBefficiency
P4. Since there is no doubt about the thermal polar cap Lx/E˙ ∼1.9×10−2 givesthepredictedvalueofP4 ∼(6±1)P
emission component, this case greatly strengthens our ar- from equation (2). It is very interesting that WES06 re-
gumentsgiven for PSRsB0943+10 andB1133+16, and the port the periodicity of (7±1)P (see Table 1 and Fig. 1).
equation (3) receivesaspectacular confirmation. It isinter- According to the model expressed by equation (3) this rel-
esting to note that the erratic radio emission detected by atively low modulation periodicity (i.e. high modulational
Weltevrede et al.(2006b) is similar to the so-called Q-mode frequency) can be interpreted as the circulation time P4. If
in PSRB0943+10. Thelow frequencyfeature inthefluctu- thisistruethenPSRB0628-28 isnotanexceptionalpulsar
ationspectra,identicaltotheoneintheorganized B-mode, atall.ItliesonthetheoreticalcurveinFig.1atexactlythe
wasfoundbyRankin&Suleymanova(2006;seetheirFig.6). rightplace.Thisalsomeansthattheobserveddriftishighly
Asgekar & Deshpande (2001;AD01 hereafter) also detected aliased in this pulsar, with P3/P being considerably lower
this feature in the 35-MHz observations of PSR B0943+10 than2. Asconcluded byWES06,thismight bethecasefor
(see theirFigs.1 and 2).This simply means that the E×B most pulsars. Therefore, all or most features in modulation
plasmadriftismaintainedinbothregular(withdriftingsub- spectrum frequencies below about 0.2 cycle/P may in fact
pulses observed) and erratic (no drifting subpulses) pulsar represent directly the E×B plasma circulation around the
emissionmodes.Seesomeadditionaldiscussioninsection4. pole rather than theapparent subpulsedrift periodicity.
X-ray pulsar radiation from polar cap heated by back-flow bombardment 5
4 CONCLUSIONS AND DISCUSSION driftmotionisalwaysperformedatthesamerateinagiven
pulsar. The problems is how to reveal this motion.
Within the partially screened gap (PSG) model of the in-
Different methods of analysis of pulsar intensity fluc-
neracceleration region in pulsarsdevelopedbyGMG03, we
tuations are sensitive to different effects. The method used
derived a simple relationship between the X-ray luminosity
WES06 has an obvious advantage of finding periodicities
L from thepolar cap heated bysparks andthecirculation
x eveninaveryweakpulsars,soitresultedinalargeincrease
timeP4 ofthespark-associateddriftdetectedinradioband, of pulsars with drifting subpulses and/or periodic intensity
not necessarily in the form of regularly drifting subpulses.
modulation.Generally,WES06canfindonlyoneperiodand
ThisrelationshipexpressesthefactthatbothE×Bdrifting
theydenotealltheperiodstheyfindbyP3,suggestingthat
rateandpolarcapheatingratearedeterminedbythesame
these are primary drift periodicities. It does not have to
value of the available potential drop. In PSRs B0943+10,
be this way at all. In fact, we suggest that at least in three
B1133+16, B0628−20 and B0654+14, which are the only
casestheirreportedvaluescorrespondtocarouselcirculation
pulsarsforwhichbothLx andP4 areknownatthemoment, timesP4.Webaseourargumentmainlyonthefactthatthey
the predicted relationship between observational quantities
satisfy nicely our empirical relationship (Eq. 3 and Fig. 1),
holdsverywell (Fig. 1 andTable 1).This suggests that the
without any obvious selection effect involved. Moreover, in
PneSrGacmceolderealtmorayneinadretehdebpeolaarrecaasponreagbiloend.eWscirtihpttihoenaobfutnhdeainnt- B1133+16thevalueofP4 =(33±3)P iscloseto(37.4+−01..44)P
detectedinthetwinpulsarB0943+10 (seePaperIformore
radiodriftingdata(WES06)andthegrowingnumberofold
detailed discussion). In B0656+14 the periodicity of about
pulsars detected in X-rays by XMM-Newton and Chandra,
20P resultsfromintensitymodulationoferraticspikyemis-
the clean prediction from this model (equation 3) will be
sion, similar to thecase of Q-modein B0943+10.
unambiguously further tested with more pulsars in the fu-
Using a concept of thecomplexity parameter a (GS00)
ture.PSRB0826-34 withP4 about14or7.5P (Gupta,Gil, correspondingtotheratioofthepolarcapsizetothespark
Kijak et al. 2004) and PSR B0834+06 with P4 about 15P, characteristic dimension, we estimated a number of sparks
will be examined in thenear future.
,N, operating in the inner accelerating regions, as well as
ForthecarouselcirculationtimeP4tobemeasurableat valuesofthescreeningparameterηforthepulsarsdiscussed
all,itrequiresastrongunevennessinthecirculatingsystem,
in this paper. In the two twin pulsars both N and η are
maybeadistinguishedgroupofadjacentsparksorevenjust
almostthesame.Isseemstrivialsinceintheapproximation
a single spark (see also scenario discussed by Gil & Sendyk we used a depends only on the P and P˙ values, which are
(2003). Moreover, it requires this feature to persist much
close to each other for these two pulsars. However, N can
longer than the circulation time. Such favorable conditions
also be found from the ratio of observed values of P4 and
donotoccurfrequentlyinpulsarsandthereforedirectmea-
P3,andbothestimatesareconsistentwith eachother.PSR
surements of P4 are very rare. In principle, in a clean case, B0628-28 seems quite similar to the twin pulsars, while in
usingthefluctuationspectraanalysis,oneshouldbeableto
PSR B0656+14 the number of sparks is 4 times greater,
detect the primary feature P3, reflecting the phase modu- andthescreeningparameterisquitehigh(correspondingto
lation of regularly drifting subpulses, flanked by two sym-
about 75 % of the vacuum potential drop). This is a result
metrical features corresponding to slower amplitude modu- of relatively low P (large polar cap) and unusually high P˙.
lation associated with carousel circulation. PSR B0943+10
Thus, either the actual number of sparks is really that big
was the first pulsar to show such a model behavior (DR99)
inthispulsar,ortheapproximationofGS00isnotgoodfor
and PSR B0834+06 was the second one, as demonstrated
suchanon-typicalpulsar.Itisnotdifficulttolowerthevalue
by Asgekar & Deshpande (2005). The latter authors have
ofthecomplexityparameterandacorrespondingnumberof
alsofoundadirectlongperiodcirculationalfeaturesinboth sparks (N =πa) by a factor of 2−3, by considering larger
pulsars. For the B0943+10 they found it in their 35-MHz
radii of curvature of the actual surface magnetic field lines,
observations (AD01). In the case of B0834+06 Asgekar &
oreventheinverseComptonscatteringinsteadofcurvature
Deshpande(2005)foundanoccasionalsequenceof64pulses
radiation as seed photons for the sparking discharges (see
withmuchweakerfrequencymodulation(presentintherest
Zhang,Harding&Muslimov2000andGil&Melikidze2002
oftheirdata)butwithstronglongperiodfeatureassociated
for some details).
withtheamplitudemodulationduetothecirculationofone
or few sparks (see their Fig. 3)3 Most interestingly, how-
ever, Rankin & Suleymanova (2006) were able to detect a
ACKNOWLEDGMENTS
longperiodcirculationalfeatureP4 inthesocalled Q-mode
erratic emission mode in B0943+10. This apparently first Wethank Geoff Wright for valuable criticism. We acknowl-
detectionoftheQ-modecirculation timeisveryimportant. edgethesupportofthePolishStateCommitteeforscientific
Indeed,thisfactandothercasesdiscussedinthisparagraph, research under the grant 1 P03D 029 26. GM was partially
strongly suggest that no matter thedegree of theorganiza- supported by Georgian NSFgrant ST06/4-096.
tion of spark plasma filaments at the polar cap, the E×B
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