Instabilities nulls and sub-pulse drift in radio pulsars

0.23 MB·English

by  P. B. Jones

Mon.Not.R.Astron.Soc.000,1–12() Printed1February2011 (MNLATEXstylefilev2.2)
Instabilities nulls and subpulse drift in radio pulsars
P. B. Jones⋆
1 Department of Physics, Universityof Oxford, DenysWilkinson Building,
1 Keble Road, Oxford OX13RH
0
2

n
a
J ABSTRACT

1
3 This paper continues a previous study of neutron stars with positive polar-cap

corotationalchargedensityinwhichfreeemissionofionsmaintainsthesurfaceelectric-
E] fieldboundaryconditionE·B=0.The compositionofthe acceleratedplasmaonany
subset of open magnetic flux-lines above the polar cap alternates between two states;
H
eitherprotonsorpositronsandions,ofwhichtheprotonstatecannotsupportelectron-
.
h positronpaircreationathigheraltitudes.Thetwostatescoexistatanyinstantoftime
p above different moving elements of area on the polar cap and provide a physically
- consistentbasisforadescriptionofpulsenullsandsubpulsedrift.Inthelattercase,it
ro isshownthatthebandseparationP3isdeterminednotbytheE×Bdriftvelocity,asis
t generally assumed, but by the diffusion time for protons produced in reverse-electron
s
showers to reach the region of the atmosphere from which they are accelerated. An
a
[ initial comparisonis made with the survey of subpulse drift published by Weltevrede

et al.
1
v Key words:
8 pulsars: general -stars: neutron - plasma - instablities
1
9
5
.
1 1 INTRODUCTION withinitisnotsufficientlyrapidtohinderthechargemove-
0
ment necessary to maintain the boundary condition. Thus
1 It is nowgenerally assumed that electron-positron pair cre-
there are three possible cases depending, firstly, on the re-
1 ation immediately above the magnetic poles is essential lationbetweenrotation spinΩandBandhencethesignof
v: for coherent radio emission in pulsars. Muslimov & Tsy- the polar-cap Goldreich-Julian charge density ρ , and sec-
Xi greacnog(n1i9t9io2n) hofavtehemsaigdneifiacnanimcepoorftathnet Lcoenntsrei-bTuhtiirorninignetffheecitr o(in)dΩly·oBn >the0,nρeut<ron0-,stEar s=ur0fawceitvhaleuleectorfonEka.c0cTelheeryatiaorne;:
ar ianlsothtehegepnaepreartioofnMoufsalidmeoqvua&telHyarsdtrionngg19e9le7c)t.riIct ifiserldems a(sreke- b(iai)ryΩon·Bacc<ele0r,atρi00on>o0r,(iEiikk) Ω=·0Bw<ith0,poρsit>ron0 aannddEpos6=itiv0e.
able, and must be almost unique,that a general-relativistic 0 k
Throughout thispaper, for brevity,theseconditions will be
effect changes the order of magnitude of a field component
referred to as cases (i), (ii) or (iii). There is nofurther pos-
in asystemthat canbedescribed,ostensibly,perfectly well
siblecase because freeemission ofelectrons is always possi-
in Euclidean space as in the paper of Goldreich & Julian
ble at neutron-star surface temperatures. A previous paper
(1969).
(Jones 2010a, hereafter PaperI) reported some preliminary
Atleastinitially,findingtheaccelerationfieldE which
k work on therelation between these sets of boundary condi-
is the component locally parallel with the magnetic flux
tions and observed phenomena, in particular, the existence
density B, can be regarded as a problem in electrostatics
ofpulsenullsinasignificantfractionofradiopulsars.There
within a volume in the corotating frame of reference hav-
seems to be no reason why neutron stars satisfying (i), (ii)
ingwell-definedboundaryconditions.Thisisdefinedbythe
and even exceptionally (iii) should not exist and it is the
neutron-star surface and the surface separating open from
purposeofthepresentpapertocontinuetheattempttosee
closed magnetic flux lines and is assumed to have the elec-
iffeaturesofthepredictedplasmaaccelerationinthesecases
tric potential boundary condition Φ=0. It is possible that
correlate with observed properties of subsets of the pulsar
this surface is actually a current sheet of some complexity
population.
(see, for example, Arons 2010), but we shall assume that,
near the neutron-star surface, the time-dependence of E Therecanbelittledoubtthatinthiscontextnullsand
k subpulse drift are important phenomena. Although there
must be reservations about assigning undue weight to the
propertiesofasingle neutronstar,nullsintheisolated pul-
⋆ E-mail:[email protected] sar PSR B1931+24 are informative. Kramer et al (2006)
2 P. B. Jones
foundthatspin-downintheon-stateisapproximatelytwice the formation of organized subpulses. It was shown in Pa-
as fast as in the off-state. It is difficult to see, in a pulsar per I that short time-scale instability in the composition of
of this age (> 106 yr), how the geometrical shape of the accelerated plasma exists in case (ii). In this paper, it is
accelerationvolumeorthemagnitudeofthecurrentdensity shown to be a plausible mechanism for subpulse formation
J within it can change bya factor of thisorder in less than and drift.
the rotation period P. Time-varying fields near the light Undertheassumptionofanactualdipolefield,paircre-
cylinder can further accelerate ultra-relativistic particles of ationispossible,inprinciple,inallobservedpulsarsthrough
both signs and therefore the most obvious explanation for the inverse Compton scattering (ICS) of polar-cap photons
thechangeinspin-downtorqueisthattheparticleandfield by accelerated electrons or positrons (Hibschman & Arons
components of the magnetospheric momentum density and 2001,alsoseeFig.1ofHarding&Muslimov2002),butcur-
stress tensor near the light cylinder differ between on and vatureradiation (CR) can produce pairs only in those with
off-states. A cessation of pair creation during nulls would high values of B P−2. However, there is a problem, noted
d
greatly change the charged particle number density even by Harding & Muslimov, in the formation of a dense pair
thoughJremainsessentiallyunchanged.Inthisconnection, plasmabecausethenumberofICSpairsformedperprimary
it is interesting that a recent paper by Lyne et al (2010) electron or positron accelerated can be smaller than unity
suggestsquasi-periodicswitchingbetweentwodifferentspin- (see also Fig. 8 of Hibschman & Arons). They suggest that
downratesastheoriginofthepeculiartimingresidualsseen thepairdensityrequiredforcoherentradioemissionmaybe
in many pulsars. far smaller than previously thought. Although higher mul-
The presence of nulls might be thought of as evidence tipole field components may be present in most pulsars, so
thatapulsarisinthefinalphaseofevolutionpriortocom- increasing flux-linecurvature,the existence of this problem
plete cessation of radio emission. However, we suggest that must remain a matter of concern perhaps even leading to
this view is not consistent with a specific feature of the 63 doubtsabouttheroleofpairplasmaasthesourceofcoher-
nullingpulsarslistedinTables1and2ofthepaperbyWang, ent radio emission.
Manchester&Johnston(2007).Themaximumpotentialdif- The existence of solutions of the electrostatic problem
ference Φmax available for acceleration at the polar cap is under boundary conditions (ii) or (iii) is no more than a
proportional to BdP−2, in which Bd is the effective dipole preliminarybecauseweareconcernedinthispaperwiththe
field inferred from the spin-down rate. It is approximately presenceofinstabilitieswhicharisefromthereverseelectron
independent of the ratio of B to the actual polar-cap field flow at the polar cap. It might be questioned whether or
d
B. The distribution of this quantity for all radio pulsars in not the charge density on the surface separating open from
theATNFcatalogue(Manchesteretal2005)hasarelatively closed magnetic flux-lines needed for the condition Φ = 0
sharpcut-offat2.2×1011 Gs−2,equivalenttotheexistence can be maintained in the presence of instability. But we
of a well-defined death line in the distribution of Bd versus shall find that, even at short time-scales, instability princi-
P.It might beexpected thatnulls should beseen only as a pally changes the nature of the particles accelerated rather
pulsar rotation slows so that it approaches this value. But than the current density J and the acceleration field. The
theformofthedistributionofBdP−2forthenullingpulsars instabilities considered here are not, of course, a feature of
listed by Wang et al is broadly indistinguishable from that case (i) in which electrons aretheprimary componentof J.
ofthewholeATNFcatalogueandinsteadisconsistentwith Establishing instability in this case is a purely electromag-
nullsbeingalong-termpropertyofacertainsub-setofradio netic problem which has been considered by Levinson et al
pulsars.Itis,ofcoursejustpossiblethattheobservedsharp (2005), Melrose & Luo (2009) and Reville & Kirk (2010).
cut-off is itself a statistical fluctuation and that individual Non-stationary flow in case (iii) has been investigated by
pulsar deaths actually occur at values of BdP−2 through- Timokhin (2010).
out the whole distribution because some other variable is Thepropertiesofthecondensedmattersurfacearenat-
involved, such as flux-line curvature. But we shall discount urally important in cases (ii) and (iii), in particular, the
thispossibilityandinthepresentpaperadopttheviewthat production of protons by the reverse flux of electrons de-
pulsarswithboundarycondition(ii)and,exceptionally,(iii) scribed previously (Paper I; see also Jones 1981). This is a
form the sub-set that null. characteristic of electromagnetic showers that is usually of
A recent large-scale survey by Weltevrede, Edwards & littlepracticalsignificance.Theformoftheshowerdepends
Stappers(2006)hasrevealedthatsubpulsedriftisacommon principally onelectron-photon interactions butshowerpho-
phenomenon. Under the assumption that electron-positron tonsalsointeractdirectlywithnuclei.Thecross-section isa
pairs are the source of the coherent radio emission, it im- maximum with the excitation of the giant dipole resonance
plies that compact zones of pair creation exist and movein (GDR), a broad collective state, at a photon momentum
anorganizedwaywithintheopenmagnetosphereareaofthe k≈40mc.(Electroproduction cross-sections aresmallerby
polar cap. This motion has been assumed to be an E×B a factor of the order of the fine-structure constant and can
driftvelocityunderthecase(iii)surfaceelectricfieldbound- be neglected.) Photon track length per unit interval of k in
ary condition, following the original paper of Ruderman & ahigh-energyshoweris∝k−2 sothattoagoodapproxima-
Sutherland (1975). There have been many later papers on tion, photoproduction of baryons can be assumed to occur
thissubjectandwerefertoGil,Melikidze&Geppert(2003) entirelythroughGDRformation.Knowncross-sections and
forrecentdevelopmentswhichhavesoughttorefinethecal- electromagnetic shower theory allow the calculation of pro-
culation of the drift velocity. The problem is that the case duction rates, and we define W as the number of protons
p
(iii) boundary condition is implausible as a common prop- produced per unit primary shower energy. Protons are ini-
ertyofthepulsarpopulationandthatcases(i)and(ii)have tiallyoftheorderofafewMeVbutareveryrapidlymoder-
not hitherto provided any immediate and obvious basis for atedtothermalenergieswithoutfurtherstronginteraction,
Nulls and subpulse drift 3
and then diffuse to the surface with a time delay that is of defined here in terms of the zero-field Bethe-Heitler
prime significance for the stability of plasma acceleration. bremsstrahlungcross-section withscreeningfactormodified
ForfurtherdetailswerefertoPaperI,alsotoJones(2010b) for theneutron-star surface density (see Paper I).
forthecross-sectionsathighvaluesofBforprocessesofsec- The critical temperature above which the ion ther-
ondorderinelectron-photoncoupling.Intheearlystagesof mal emission rate is high enough to maintain the case (ii)
acceleration,partially-ionizedatomsinteractwiththepolar- boundary condition is related to the cohesive energy E by
c
cap blackbody radiation field and this is the source of the k T ≈ 0.025E (see the discussion of atmospheric prop-
B c c
reverse-electron flux that is considered here. It has proved erties in Section 3). Cohesive energies have been calcu-
convenient to combine rates for this process with values of lated by Medin & Lai as functions of B. For Z = 26 and
W by defining, for a particular pulsar, the parameter K B = 10, their value is in good agreement with that ob-
p 12
which is equal to the number of protons produced per unit tained by Jones (1985) using a different representation of
nuclear charge accelerated. thethree-dimensional condensed matter state. In the inter-
Instabilities in solutions of the time-independent elec- val10<B <100,theirvaluescanbefittedbytheexpres-
12
trostatic problem referred to above exist as a consequence sionsE =0.016B1.2 keVfor Z =6and E =0.16B0.7 keV
c 12 c 12
of proton formation and we proposethat they are thebasis for Z =26 giving,
forthenullingphenomenonandforsubpulseformation and T = 4.6×103B1.2 K, Z =6
drift. The complexity that arises is unfortunate because it c 12
= 4.6×104B0.7 K, Z =26. (3)
limits what can be achieved in terms of a physical theory 12
of the acceleration process. Thus we shall be able to ex- These are to beseen in relation to other temperatures that
pose the general properties attaching to cases (ii) and (iii) aresignificant.PaperIcontainedacalculationoftherateof
butarenotalwaysabletogivequantitativepredictions.We proton formation in theelectromagnetic showers formed by
shall show in Section 2 that general consideration of polar- reverseelectronsincidentonthepolarcap.Fromtheenergy
cap parameters rule out the possibility that the relatively fluxneededtoproduceaprotoncurrentdensityJp =ρ c,we
0
numeroussubsetsofpulsarsthatexhibitnullsandsubpulse can infer a maximum steady-state polar-cap temperature,
drift belong to case (iii). Therefore, sections 3 and 4 of this
1/4
paper are restricted to further consideration of case (ii) for T¯= T4 + (−Bcosψ)(1−κ) K. (4)
whichshorttime-scaleinstabilityinplasmaaccelerationhas (cid:18) res PσeWp (cid:19)
beendescribedpreviouslyinPaperI.Itspropertiesaresum-
Inthisexpression,W isthenumberofprotonsproducedper
p
marized in Section 3. The previous treatment of medium
unitprimaryshowerenergy.Wecanapproximateitinitially
time-scales was, however, inadequate and contained an er- by WBH which was obtained in Paper I by using the zero-
ror.TheappropriateanalysisisgivenhereinSection4.The p
fieldBethe-Heitlerpaircreationcross-sectionwithscreening
relations between these instabilities and theobserved prop-
modified for the condensed matter density at the neutron-
erties of nullsand of subpulsedrift are described in Section
star surface. Its values, given in Table 1 of that paper, can
5 and,in particular, therelation beween thesubpulseband besummarizedconvenientlyintheintervalsB =101−102
12
separationperiodicityP andprotondiffusiontimeisgiven.
3 and Z =10−26 by theexpression
WBH =3.9×10−4hZi−0.6B0.2 (mc2)−1, (5)
p sm 12
in which the nuclear charge is the average in the region
2 POLAR-CAP PARAMETERS of the shower maximum. The angle ψ is that between Ω
and B. The general-relativistic correction contained in the
Many of the polar-cap properties and parameters that will
surface value of the Goldreich- Julian charge density ρ is
berequiredfortheargumentsofSections3-5havebeengiven 0
κ (see Muslimov & Harding 1997), and σ is Stefan’s con-
previouslyinPaperI.Thepropertiesofthepolarcapatmo-
stant. Equation (4) also contains a further temperature,
sphereareofparticularimportanceandwillbefurthercon-
T , which is the polar-cap temperature in the absence of
sideredinSection3.Butitwillbeconvenienttosummarize res
any reverse electron or photon energy flux (approximately
theremainder here with some additions.
theobservedwhole-surfacetemperaturecorrectedtothelo-
The most basic parameter is the ion number density
cal proper frame). The presence of T assumes that there
N of the condensed matter at zero pressure. The magnetic res
is a constant heat flow to the surface driven by the very
dipole fields B inferred from pulsar spin-down rates vary
d
much higher temperature of the inner crust. With T =0
over about five orders of magnitude, but the median value res
for the 63 nulling pulsars listed by Wang et al is 2.8×1012 and κ=0.15, we find,
G, which is significantly larger than for the whole ATNF −cosψ 1/4
T =5.1×105hZi0.15B0.2 K (6)
catalogue.Itisalsopossiblethattheactualpolar-capfieldB max sm 12 P
(cid:16) (cid:17)
islargerthanB .Forthesereasons,weadopttheexpression,
d Equating this with T allows us to estimate the minimum
c
N =2.6×1026Z−0.7B1.2 cm−3, (1) polar-cap magnetic flux densities necessary to sustain the
12 case (iii) boundarycondition E 6=0. These are,
k
fittedtovaluescomputedbyMedin&Lai(2006),primarily
forB >10,whereB isthemagneticfluxdensityinunits −cosψ 1/4
12 12 B = 181 Z =6
of1012 GandZ istheatomicnumber.Theconvenientunit 12 (cid:16) P (cid:17)
of depth below thesurface at z=0 is theradiation length, −cosψ 1/2
= 327 Z =26. (7)
P
−1 (cid:16) (cid:17)
lr =1.66Z−1.3B1−21.2 ln 12Z1/2B1−21/2 cm (2) Comparison with the median value of Bd12 = 2.8 for the
(cid:16) (cid:16) (cid:17)(cid:17)
4 P. B. Jones
Table 1. The table gives values of Wp for high magnetic flux in Tables 2 and 4 of Jones (2010b), and the high magnetic
densitiesB>Bc andnuclearchargeZ.Theeffectivetotalcross- conversion transition rates reduce GDR-band photon track
sections σP for pair creation in the GDR photon energy band lengthintheshower.ThereducedWp valuesincreaseTmax.
arefortransversemomentabelowthelowestmagneticconversion Owing to this complexity at B ∼ Bc there are inevitable
threshold at k⊥ = 2 mc and are in units of barns. They are uncertainties in our estimates of Wp, but we believe that
estimates obtained from Table 3 of Jones (2010b) by averaging they do not seriously invalidate the estimates of the min-
over photon polarization and over photon transverse momenta imum polar-cap magnetic flux densities needed to support
k⊥=1.0and1.5mc,andareequivalent toameanfreepathfor thecase(iii) boundarycondition and ourconclusion thatit
Coulombpaircreationlengthened byafactor ηp comparedwith canexistonlyinaverysmallsubsetofpulsars.Ourconclu-
that for the modified zero-field Bethe-Heitler cross-section. The
sions drawn from equation (3) are also independent of the
Landau-Pomeranchuk-MigdaleffectisnotsignificantintheGDR
spectrum of the reverse electron-photon flux because W is
photonenergyband p
almost completely independent of primary shower energy
Z BBc−1 σP ηp Wp provided that is large compared with theGDR energy.
bn (mc2)−1 Thereappeartobenopublishedestimatesofthemelt-
ing temperature of condensed matter that are specific to
10 1 0.27 1.14 2.3×10−4 very high magnetic fields. Consequently, we are obliged to
3 0.071 2.95 3.4×10−4
adoptthestandardone-componentCoulombplasmaexpres-
10 0.017 6.3 3.6×10−4
sion (see, for example, Slattery, Doolen & DeWitt 1980)
18 1 0.89 1.35 1.8×10−4
3 0.23 3.8 3.2×10−4 which, with equation (1), gives,
10 0.055 9.7 3.8×10−4 T =1.0×104Z2Z−0.23B0.4 K, (8)
26 1 1.86 1.45 1.3×10−4 m v 12
3 0.48 4.2 2.7×10−4 intermsofaneffectivevalencechargeZv.Thislatterparam-
10 0.11 11.3 3.4×10−4 eterrepresentsthefactthatthedeeply-boundLandaustates
certainlydonotparticipatesignificantlyinthemeltingtran-
sition, but its estimation at higher values of Z is quite dif-
63 nulling pulsars listed byWanget al shows that case (iii) ficult. It is possible that the work of Potekhin, Chabrier &
canbewidelyrealizedonlywithimplausiblylargedeviations Yakovlev(1997;seeFig.1)couldprovideguidancealthough
from a central dipole field. it is at zero field and was directed toward a different prob-
However, the screening-modified zero-field Bethe- lem. On this basis, we assume for Z = 26 a value in the
Heitler pair creation cross-section is not obviously valid intervalZv =10−15.Inatypicalpolar-capfield,B12 =10,
at magnetic flux densities of the order of the critical field the melting temperature is as low as Tm = 6×105 K for
Bc = 4.41 ×1013 G. Thus we have been obliged to cal- Zv = Z = 6 but exceeds 106 K for Z = 26. Thus the state
culate the second-order bremsstrahlung and pair creation of the polar cap may be a sequence of melting and solidifi-
cross-sections using Landau function solutions of the Dirac cations. The order of magnitude of the shear modulus is a
equation. The photoproduction of protons by giant-dipole further source of complexity. The standard (zero-field) ex-
resonance (GDR) decay is determined by the total photon pression for a body-centred cubic lattice (Fuchs 1936) can
track length in the GDR band, centred on a momentum beadapted, with equation (1), togive,
k = 40 mc, which occurs almost entirely in the late stages µ=1.1×1016Z2Z−0.93B1.6 erg cm−3. (9)
of shower development. The track length at these energies v 12
is limited by Coulomb and magnetic pair creation, also by However,thepolar-capgravitationalconstantisg≈2×1014
Compton scattering the effect of which is almost always to cms−2 sothatanydensityinversionmaywellinduceaform
scatterthephotonsothatitstransversemomentumcompo- of Rayleigh-Taylor instability.
nent k (perpendicular to the field) exceeds the threshold The final condensed-matter parameter that is impor-
⊥
formagnetic conversion toelectron-positron pairs.Werefer tant is the thermal conductivity parallel with B, which is
toSections2and3ofPaperIforamoredetaileddescription extremely high (see Potekhin 1999). Thermal energy is de-
oftheseprocesses. ApproximatevaluesoftheCoulombpair posited at shower maxima a distance zp below the surface
creationcross-sectionatmagneticfieldsB >B aregivenin z=0,whichisdefinedasthatseparatingcondensedmatter
c
Table 3 of Jones (2010b) and are thebasis for thevalues of from the atmosphere, and is then dissipated as blackbody
WpgivenhereinTable1.Thesearenoteasilysummarizedas radiation from the polar cap. The value of zp in the high-
asimpleexpression analogouswithequation(5).Thecross- density condensed matter of the neutron-star surface de-
sectionatB=10B isatleastanorderofmagnitudesmaller pendsonshowerenergyowingtotheLandau-Pomeranchuk-
c
the the modified zero-field Bethe-Heitler cross-section but Migdal (LPM) effect but for the order of magnitude of the
theeffectonWpisnotlargebecausethephotontracklength characteristic time we can assume −zp ∼ 10lr ≈ 10−3 cm
inthisregionislimitedbyComptonscattering.Substitution usingtheradiationlengthslr givenbyequation(2)orinTa-
into equation (4) gives values of T that typically are re- ble 1 of Paper I. The characteristic time for shower energy
max
duced by a factor of approximately 0.9 from those of equa- input toproduce a surface-temperature fluctuation is then,
tion (6). But the complexity of the second-order processes Cz2
at B ∼ B is such that we have not reconsidered shower t = p ≈10−9 s, (10)
c cond 2λ
development and,specifically, havenot allowed for thepro- k
duction by cyclotron emission or Coulomb bremsstrahlung in which typical values of the parameters are the specific
of photons with k above the thresholds for magnetic con- heat C =1.0×1012 ergcm−3 K−1 and thelongitudinal co-
⊥
version. This becomes significant at B = 10B , as shown efficentofthermalconductivityλ =6×1014 ergcm−1 s−1
c k
Nulls and subpulse drift 5
K−1. But for a surface temperature of 106 K, the internal the order of 1s, the temperature and depth of the atmo-
temperaturegradientneededtoconducttheradiatedenergy sphere can change appreciably but there is no doubt that
flux is extremely small, of the order of 105 K cm−1, within it is always in local thermodynamic equilibrium. The pro-
thecondensedmatteratz<0.Ineffect,heatismoreeasily cessesofprotonformation byGDRdecayfollowed bydiffu-
conductedtogreaterdepthsthanradiatedfromthesurface. siontothesurfaceweredescribedinPaperI,whichassumed
Consequently, the Green function G(z,z ,t) giving the in- averythinatmosphere.Therefore,weneedtoconsiderhere,
p
ternaltemperaturedistributionmustbealmostindependent in more detail, diffusion in the atmosphere 0<z<z . The
1
ofz andveryclosetothatforzerotemperaturegradientat numberofprotonsisverymanyordersofmagnitudesmaller
p
thesurfacez=0.ThusashowerheatinputHδ(t)produces than that of ions so that the properties of the atmosphere,
a temperature distribution within the condensed matter at its scale height and equilibrium electric field given by the
z60, electrical neutrality condition, are determined solely by the
latter. With the assumption of a single ion component we
H −Cz2
T(z,t)=HG(z,t)≈ exp (11) can determine theequilibrium electric field within the LTE
(πCλkt)1/2 (cid:18) 4λkt (cid:19) atmosphereinthepresenceofagravitational acceleration g
and find that theproton potential energy is,
It is asymptotically ∝t−1/2 so that the polar-cap tempera-
turearisingfromasequenceofheatingevents,eachproduc- A
1− m gz=αm gz (12)
ing a maximum temperature Tmax, has fluctuations away (cid:18) Z˜+1(cid:19) p p
fromT¯whosemagnitudeisafunctionofthetime-scalecon-
cerned. Further discussion of this is deferred until our con- for ions of charge Z˜ and mass number A, and that this
sideration of observed polar-cap blackbody temperature in transports protons to z > z1 from which region they are
Section 5. accelerated to relativistic energies in preference totheions.
Proton diffusion at a rate greater than is needed for a cur-
rentdensityJp =ρ cthereforecutsoffionacceleration and
0
produces a thin electrically neutral proton atmosphere at
3 SHORT TIME-SCALE INSTABILITY z>z .Thereappearstobenoreasonwhythisatmospheric
1
structureshould bedisturbed by turbulentmixing.
Undertheassumptionthatthecase(ii)boundarycondition The proton average potential energy at z < 0 is close
is satisfied, the neutron-star surface has an atmosphere in to m gz but the jump bias, m ga /k T for ion separation
p p s B
local thermodynamic equilibrium (LTE) with approximate a , is too small to be significant because m gz /k T ≪ 1.
scaleheightlA =(Z˜+1)kBT/Mg∼10−1cmattemperature Tsheatmosphericprotondensityatz =0isnpecepssarBilysome
T = 106 K, where Z˜ and M are the mean ion charge and ordersofmagnitudesmallerthanthedensityatz <0owing
mass of the partially ionized atom. The expression for the tothedensitydiscontinuityatthesurface.Thusdiffusionto
chemicalpotentialofanidealBoltzmann gasgivestheLTE the surface at z = 0 is little changed by the presence of
atmospheric numberdensity at z=0, an atmosphere that is not necessarily very thin. Movement
Mk T 3/2 of the protons through the ion atmosphere is effected by
NA(0)=(cid:16) 2π¯hB2 (cid:17) e−Ec/kBT, tnhuemcbheermdiecnaslitpyo.tAentthiailghgrdaedniesnittiews,haicshinisthaefsuonlcidtioant zof<io0n,
for atmospheric temperatures such that NA(0) ≪ N. Its thisisdeterminedprincipally byentropybutasthedensity
order of magnitude is NA(0) ≈ 1032exp(−Ec/kBT) cm−3 reduces,thepotentialenergygivenbyequation(12)becomes
for M = 56mp. We estimate the critical temperature Tc the more important factor. In effect, the motion changes
which must not be exceeded if the case (iii) boundary con- from a random walk to a drift velocity. The dividing ion
dition is to be valid by equating the flux of ions in such density is given by the condition λ = N−1/3 and is ≈
R A
an LTE atmosphere that are incident on the neutron-star 4×1022 cm−3, below which it is appropriate to define a
surface with the ion flux needed to give an open magneto- driftvelocitydeterminedbythelocalvalueofthemeanfree
spherecurrentdensityequaltoρ0c.ThisgivesNA(0)≈1014 pathλR derivedfrom thetotal cross-section for Rutherford
cm−3 and kBTc = 0.025Ec. But the extent to which the back-scattering,
atmosphere can be described as thin is very temperature-
m 1/2
doredpeerndofen1t0.23Thcmus−f3oranTd=the2TwchaonledaZtm=os2p6h,erNeAc(o0n)taisinosfiothnes v¯≈−αgλR(cid:16)kBpT(cid:17) , (13)
equivalenttoabout10−1l .AtadensityN (0)>1023−1024 in which,
r A
cm−3,theBoltzmanngasestimateofl assumedhereisun-
A 1 k T 2
reliableanditbecomesnecessarytoallowfortheinteraction λ = B . (14)
of protons with the ion atmosphere. R πNA (cid:16)Z˜e2(cid:17)
Ifinstabilitiesinplasmaaccelerationwithtime-scalesas The drift velocity is v¯ ≈ 2.6 cm s−1 at this density and
shortas∼10−4 sareconsidered,thetemperatureatz=0is T = 106K. The consequence for a T = 2T atmosphere is
c
constantapartfromverysmallfluctuations,asshownbythe thatthediffusiontimetoz=z isincreased,bysomeorders
1
GreenfunctiongivenattheendofthepreviousSection.Con- of magnitude, compared with the values assumed in Paper
sequently,thetemperaturedistributionandnumberdensity I which were for diffusion through the condensed matter
of ions in theatmosphere are also constant in time and the only.Also,thechangefromrandomwalktodriftvelocityin
atmosphereisinlocalthermodynamicequilibriuminthein- the atmosphere has a significant effect on the distribution
terval0<z <z ,wherez isthetopoftheionatmosphere, of diffusion times because it removes the long tail in the
1 1
defined as the surface of last scattering. For time-scales of distribution that is a feature of random walks.
6 P. B. Jones
Some effort has been made here to confirm that there GeV,islessimportantintheatmospherewithinwhichthere
isoutwardproton diffusionbecausetheprocessisofcrucial will be some GDR formation. But we have been unable to
importance. It was shown in Paper I that plasma accelera- make a satisfactory quantitative estimate of pair formation
tionisunstable,consistingofalternateburstsofprotonand through this process.
ionacceleration, withtime-scaledeterminedbythetimefor
proton diffusion from z = z to z = z . The polar-cap cur-
p 1
rentdensityJisessentiallytime-independentforrelativistic
flow (as is the charge density) so that there is no fluctua-
tion in polar-cap electric field other than that arising from 4 MEDIUM TIME-SCALE INSTABILITY
thereverseflowofphoto-electronsfromacceleratedions(see
Instability on time-scales some orders of magnitude longer
Section5).Theinstabilityisinthenatureoftheplasmaac-
thanthoseof Section3isalsoof interestandcan appearas
celerated and remains adequately described by theanalysis
a fluctuation in the charge of nuclei reaching the polar-cap
given in Paper I which will not berepeated here.
surface which we shall define here to be always at z = 0
The maximum acceleration potential difference, given
although it may move with respect to coordinates fixed at
a current density J = ρ c, is almost entirely dependent on
0
thecentreofthestar.IonsofinitialchargeZ moveupward
the Muslimov & Tsygan (1992) general-relativistic correc- a
through the region of the shower maxima at z and, with
tion. There is aseparate contribution arising from ion iner- p
nuclearchargereducedtoZ byGDRformation anddecay,
tia (Michel 1974) but it is important only at very low alti- b
enter the atmosphere at z > 0. We wish to find if there
tudes. Neglecting this, the maximum potential difference is
areconditionsunderwhichthelocalaveragenuclearcharge
approximately
Z (z) fails to be a time-independent function of position
0
|Φ |≈ 2π2κBdR3 ≈7×1012κBd12 volts (15) with limits Za > Z0(z) > Zb. The work of this Section
max c2P2 P2 replaces that of Section 4.1 in Paper I which contains an
error.
and, unless restricted by pair creation at lower altitudes, is
The longer time-scales here enable us to assume the
reached at an altitude z of the order of the neutron-star
protonandioncurrentdensitiesJp andJz arethetimeav-
radius Rwhich is roughly two orders larger than thepolar-
erages of those described in the previous Section. There is,
cap radius. We refer to Harding & Muslimov (2001, 2002)
of course, a distribution of discrete nuclear charges in the
foracompleteaccount.Weassumeherethatforthiscurrent
shower region, but we shall work in terms of the average
density,spontaneouspaircreation bycurvatureradiation is
Z(z,t), the corresponding number density N(z,t) given by
notpossible. Then theprimarysourceofanypositron com-
equation (1), and the velocity v(z,t) with which nuclei ap-
ponent in J can only be the reverse flow of photo-electrons
proach thepolar-cap surface at z =0. Proton formation by
fromaccelerated partially-ionized atomsasdiscussedinPa-
GDR decay occurs predominantly in thevery late stages of
per I. But there is an essential difference here in that pro-
shower development, as explained in Section 1, and so we
tons in the very low density region at z ∼ z are almost
1 shall assume that it is confined within limits z and z and
completely ionized (here,werefertoFig.1ofPotekhinet al a b
is defined bythenormalized function g (z),
2006) so that both track length and energy flux of reverse- p
flowelectronsfromphoto-ionizedhydrogenatomsarenegli- zb
g (z)dz=1. (16)
giblysmall.Thuspositronproductioninanyintervalofpro- p
Z
tonaccelerationisnegligible.Thisisalsotrueforionsoflow za
atomicnumberZ ∼4−5whicharecompletelyionized,but The physical basis for our study of the system is that the
forhigherZ,thereversefluxofinwardacceleratedelectrons totalnumbersof nucleiandof protons(boundorunbound)
produces polar-cap ICS photons, as would outward accel- are conserved. Because both neutrons and protons are pro-
erated electrons in case (i). Pairs are produced by photons duced in GDR decay, we can make the approximation of
that are scattered to transverse momenta above the mag- neglecting the effect of β-transitions. Therefore, the conti-
netic conversion thresholds. The only difference is that the nuityequations are,
photonsareinwardmoving.ButevenifspontaneousCRpair
∂N ∂(Nv)
formationisnotpossible,positronsacceleratedoutwardwill + =0, (17)
∂t ∂z
produceCR pairs at higheraltitudes, as in case (i), though
superimposed on a fluxof ions. andwithneglectoftherelativelyshortprotondiffusiontime
Incase (i),thereversefluxofpositronsmustform pro- so that within medium time-scales all protons produced in
tons which are not accelerated but form an atmosphere at theshowerareassumedtobepromptlyacceleratedfromthe
z>0 whose equilibrium isdefinedbyvarious diffusion pro- atmosphere at z≈z1,
cessesperpendiculartothemagneticflux.Backwardmoving
∂(NZ) ∂(NZv)
photonsfrom theelectron or positron showers area further + =−g (z)Jp(t). (18)
∂t ∂z p
sourceofpaircreationineachofcases(i)-(iii).Thesearise
from the decay of residual nuclei following proton or neu- Becausethereverseelectronfluxfromphoto-electricioniza-
tron emission inGDR formation, andfrom (n,γ) reactions. tion isthesourceof theshowers, weshall findit convenient
Thosephotonsthatarenotabsorbedinthemoredensepart here, as in Paper I, to introduce the parameter K which
of theatmospherecan producepairs if theirtransversemo- is a function of the atmospheric nuclear charge and is the
menta exceed the threshold. The LPM effect, which in the numberofprotonsproducedperunitpositivenuclearcharge
high-density condensed matter at z < 0 is significant for accelerated. With the representation of equation (1) in the
electrons of more than 10 GeV or photons of more than 2 form N = CZγ, equations (17) and (18) can be combined
Nulls and subpulse drift 7
to give, intermsofwhichwecouldexpressZ asanexplicitfunction
0
zb ∂Z ∂Z of z, which would be needed to obtain numerical values for
C dz Zγ +vZγ =−Jp(t). (19) the root ω = ω +iω of equation (27). However, we are
Zza (cid:16) ∂t ∂z(cid:17) primarily concer1ned h2ere not with growth rates but with
It has theobvious time-independentsolution, the boundary between stability and instability and so shall
notproceedwithnumericalsolutionsforω.Fortunately,itis
Z −Z =KZ , (20)
a b b possible to obtain a sufficient condition for the existence of
and the time-independent velocity of nuclei is such that instability independently of the form of g . Provided −1<
p
Z0γ(z)v0(z) is independent of z. γ < 0, which is clearly satisfied, we can see directly from
A natural fluctuation away from Z0 would be of the the values of the integrand in equation (27) at the limits
form Z(z,t)=Z0+δZ(z,t) with, and from its lack of an extremumthat,
δZ =η(Za−Z0(z))eiωt, (21) zbdz((1+γ)Z −γZ ) Za−Z0 Z0γ
giving similar fluctuations in Jp, N and v, in which η is Zza 0 b (cid:16) Z0 (cid:17)(cid:18)Zbγ(cid:19)
infinitesimalandindependentofzandt.Fromequation(17) <(z −z )(Z −Z ), (29)
b a a b
we then have,
and therefore that equation (27) can be replaced by the in-
δ(Zγv)=−iωηχ(z)eiωt, (22)
equality,
where,
z −z νK
z iωτ a b >−1− e−iωτ. (30)
χ(z)=γ dz′(Z −Z )Zγ−1. (23) (cid:18) |zb| (cid:19) K+1
Z a 0 0
za Thusω satisfy theinequalities,
1,2
Substitution into equation (19) with the retention of terms
of first order in η gives, ω τ zb−za > νK eω2τsinω τ
zb ∂Z 1 (cid:18) |zb| (cid:19) K+1 1
iωηeiωt dz Zγ(Z −Z )−χ(z) 0 −
Zza (cid:16) 0 a 0 1∂z (cid:17) ω2τ(cid:18)zb|z−b|za(cid:19) < 1+ Kν+K1eω2τcosω1τ. (31)
Zγv (Z −Z )ηeiwt =− δJp(t). (24)
0 0 b a C From the first of these, we can see that the real part of
The current densities, averaged over short time-scales, are ω must satisfy ω1τ > π/2, provided the proton formation
Jz = NZv(0,t) and, from the definition of K, Jp = KJz. region in theshower is sufficiently compact that,
Thefluctuationawayfrom thetime-independentsolution is
δJ =δJp+δJz =0, which we assume to bemaintained by νK eω2τ > π zb−za . (32)
the boundary conditions on Φ. To express δJp in terms of K+1 2 (cid:18) |zb| (cid:19)
δZ werepresenttheZ-dependenceofK inthevicinityofZb Withgreatercompactness, ω1τ →π,aswould beexpected.
as a power law K = K0Zν(0,t). We find, after eliminating The second inequality gives the condition for ω2 < 0, that
thevelocity fluctuation δv(0,t), that is, fluctuation growth. At the threshold where |ω τ| ≪ 1,
2
1δJp =νZγv K δZ(0,t). (25) and for zb−za≪|zb|, it is simply νK >K+1.
C 0 0K+1 The Z-dependence of K was represented in Paper I as
an approximate power law, K = K Zν with ν = 0.85 for
Therelationshipwithequation(24)isestablishedbynoting 0
Z > 10 but with the reservation that this would certainly
that,
become invalid for an atmosphere of ions with Z ∼5 which
δZ(0,t)=δZ(zb,t−τ)=η(Za−Zb)eiω(t−τ), (26) would be almost completely ionized. Therefore, we should
anticipate quite large values of ν at small Z, sufficient to
whereτ isthetimeintervalofnuclearmovementfromz=z
b
givetheunstablebehaviourfoundhere.Stabilityclearlyde-
toz=0.Fromequations(24)-(26),theequationwhoseroot
pends on the energy flux from photo-ionization. Low fluxes
ω we require can be expressed as,
and moderate rates of proton formation in theGDR region
iω zbdzZγ−1((1+γ)Z −γZ )(Z −Z ) of the showers, such that the nuclear charge inferred from
Zbγv0(zb)Zza 0 0 b a 0 equation (20) is Zb ≈ 10, allow a stable time-independent
νK progression of nuclear charge as a function of depth below
=−(Z −Z )− (Z −Z )e−iωτ, (27)
a b K+1 a b the polar-cap surface as given by Z0(z). But larger values
of K and smaller Z lead to instability. (The value of Z is
in which the function χ(z) has been removed by partial in- b a
unimportant,provideditisnotsmall,andfororderofmag-
tegration.
nitude estimations in this paper we have assumed Z =26
The only assumptions we need make about the depth a
unless otherwise stated.)
distribution of proton formation in the late stage of shower
Unfortunately,ouranalysis of theinstability of nuclear
developmentarethatitissmalloutsidetheintervalz <z <
a movement to the polar-cap surface is limited to small fluc-
z andthatz −z issmallerthan|z |thoughnotnecessarily
b b a b tuations and does not extend to large amplitudes. But it is
sobyasmuchasanorderofmagnitude.Asuitablefunction
notdifficulttoseetheformitwouldtake.Anatmosphereof
would be,
high-Zionsattimetproducesahighreverse-electronenergy
g (z))= 2 sin2 π(z−za) , (28) fluxwhichcreatesalayeroflow-Zb ionsintheshowermax-
p zb−za (cid:18) zb−za (cid:19) imum region za <z <zb. These flow toward the surface at
8 P. B. Jones
z=0andformalow-Z atmosphereattimet+τ whichpro- sibleformsthismotionmighttakewithsomeoftheobserved
ducesalowreverse-electronenergyfluxandcorrespondingly phenomenain radio pulsars.
large values of Zb. The ions accelerated alternate between The distinction between the average pulse profile and
high and low-Z values. The basic unit of time is given by individual sub-pulses within it was noted almost immedi-
thetimetrl fortheemissionattheGoldreich-Juliancurrent ately following the discovery of pulsars (see, for example,
density of one radiation length of ions, Smith 1977). The amplitude and form of successive sub-
pulses can vary in times of the order of the rotation pe-
t =2.1×105Z−1B−1(−Psecψ)
rl 12 riod and in some pulsars, observed with higher resolution,
ln 12Z1/2B−1/2 −1 s. (33) sub-pulseshavemicro-structureof10−4−10−3 stime-scale.
(cid:16) (cid:16) 12 (cid:17)(cid:17) Therearealsomoreorganizedphenomena,andforrecentex-
Thehigh-Zintervalsaresubjecttoshorttime-scaleinstabil- tensivesurveysofthesewerefertoWangetal(2007) inthe
ity asdescribed in Section 3,but low-Z intervalshavelittle case of pulse nulls and to Weltevrede, Edwards & Stappers
or no reverse-electron flux and therefore no possibility of (2006)forsub-pulsedrift.Thereisamovetowardaconsen-
significant positron acceleration and electron-positron pair sus(Lyneet al2010) thatquasi-periodicswitchingbetween
production. magnetospheric states with different spin-down rates is the
Theinstabilityoutlinedhereisofquitesimpleform,but basis of mode-changing. These authors even suggest that it
therearecomplicatingfactorsthathavebeenmentionedear- is the source of a large component of pulsar timing noise.
lier in the Section 2 consideration of polar-cap parameters. However,itisalsotruethatthesubpulsecharacteristicsob-
Evaluations of the melting temperature discussed following served in a small number of pulsars are quite singular, but
equation (8) show that there is every possibility that the here discussion is confined to the general features of sub-
condensed matter state below the atmosphere may be liq- pulses.
uid,orasolidclosetomeltingwithahighself-diffusionrate. Giventhepropertiesofthemediumtime-scaleinstabil-
Thiswouldhavenoeffectonshorttime-scaleinstabilitybut itydescribedinSection4,itisnaturaltoassociatewithnulls
could complicate the behaviour of the system over medium thoseintervalsinwhichhigh-Z ionzoneseitherdonotexist
time-scales.ThemeltingtemperatureisZ-dependentsothat or are confined to areas of the polar cap from which radia-
a density inversion is possible, the upper layer of higher- tion produced by the plasma is not visible to the observer.
Z being either liquid or solid. In the liquid case, we must For sufficiently low values of Zb, ions are accelerated from
anticipate Rayleigh-Taylor instability, which may also exist the surface completely ionized so that there is no reverse-
in the solid case because its shear modulus (see equation electronfluxandhencenopaircreationandradioemission.
9) may not be adequate to maintain mechanical stability. The current density is little changed in the open sector of
All these processes are occurring at depths |z | ∼ 10−3 cm the magnetosphere but the absence of pair creation means
p
but over a polar cap whose radius is of the order of 104 thattheparticlecontentnearthelight cylinderisquitedif-
cm. Consequently, a further complication is that different ferent as are the components of the momentum density or
polar-cap areas are unlikelytobein phasewith each other. stress tensor on any spherical surface in this region centred
These problems are considered further, though necessarily onthestar.Itisthereforenotsurprisingthatthespin-down
in a qualitativeway, in Section 5. torque is reduced during the interval of a null, as has been
observed in PSR B1931+24 (Kramer et al 2006). Neutron
stars with small K such that Z ≈ 10 are likely to have
b
ν < 1 and so will have a steady-state progressively reduc-
ing nuclear charge Z (z), no medium time-scale instablity,
5 NULLS SUBPULSE DRIFT AND 0
andnolong-termnulls.Butthequestionsaboutmeltingand
POLAR-CAP COHERENCE
mechanical stability of thepolar cap which werementioned
The time-dependent phenomena considered in Sections 3 at the end of Section 4 remain and the whole system is so
and 4 are local and one-dimensional because both shower complex and difficult to describe in physical terms that we
depth z and atmospheric scale height are very many or- areunabletogiveusefulquantitativepredictions. However,
p
ders of magnitude smaller than the polar-cap radius. This incomplete nulls, having a low but detectable level of emis-
introducesthequestionofwhetherornotthereiscoherence sionshouldbeobserved.Itisalsounsurprisingthatnullsare
overthewhole polar-cap area. Both instabilities aredepen- observed to be not completely random (Redman & Rankin
dent on theparameter K which is a function of thesurface 2009).
nuclearcharge Z(0,t)and toa lesser extentof surface tem- To some extent and unfortunately, the same remarks
peratureandaccelerationfield.Forthisreasonalone,wesee haveto bemadeabout theeffectsof short time-scale insta-
nocaseforassumingcompletecoherenceandanticipatethat bility. For temperatures T >2Tc, the proton diffusion time
thepolarcapwedescribehaszonesofprotonandionemis- ismuchlongerthaninthecaseoftheverythinatmosphere
sionwhichcannotbestationary,thetotalareasofeachbeing assumed in Paper I but remains difficult to calculate with
determined, approximately, by the average value of K. For complete confidence. In order to describe the polar cap it
neutron stars that are unable to support spontaneous pair isnecessary tointroducecoordinates u(z) on a surface per-
production by curvature radiation, the proton zones have pendicular to B. As in Paper I, the proton current density
noreverseelectronflux,donotsupportpairproduction,and at any point u(0) can be related to the ion current density
hencemerelyproduceanacceleratedone-componentplasma through thedefinition of K,
asdescribedinSection3.ButtheionzoneswillsupportICS
t
pairproductionandsoappearasmovingsourceswithinthe Jp(u,t)+J˜p(u,t)=K dt′f (t−t′)Jz(u,t′), (34)
p
polarcap.InthisSection,weshallattempttocomparepos- Z
−∞
Nulls and subpulse drift 9
inwhichf isthenormalizedprotondiffusion-timedistribu- thepointfromwhichtheionwasaccelerated,ithasnoeffect
p
tion. Without the J˜p term, whose significance is described on ion emission, which is not temperature-dependent.
below,thiswouldbeahomogeneousVolterraequationofthe In view of the time-variation described by equations
second kind having no non-zero square-integrable solution. (34)and(35),isitpossibletoimagineorganizedratherthan
(Thetime-dependenceofKarisingfromthetemperaturede- chaotic motion of ion zones on the polar cap? An example
pendenceof theLTEioniccharge Z˜ isneglected here.) The ofchaoticmotionwouldbetheexistenceofverymanysmall
approximate expression for f given in Paper I (equation zones without obvious organization. Two simple organized
p
23)assumedarandom-walkdiffusionthroughthecondensed caseswouldbemotionalongaslotofconstantrotationallat-
matter at z <0 and so is not valid for an atmosphere with itudeand circular motion at constant u(0). WiththeDesh-
properties given by equations (13) and (14). For T > 2T , pande&Rankin(1999)analysisofPSRB0943+10inmind,
c
most of the diffusion time is in the drift-velocity phase, in we consider circular motion. Equations (34) and (35) are
whichcasethetimedistributionwouldbemoresuitablyap- local in u and contain no information that can determine
proximated by a normalized gaussian function centred at an ion zone movement velocity u˙. The quantities that are
t−τ or, in the limit, by f (t−t′) = δ(t−t′−τ ), where essentiallyconstantareKand,apartfromtheeffectofvary-
p p p
τ is derived from equation (13) of Section 3. The quantity ing LTE atmospheric temperature, τ . Thus thecirculation
p p
J˜p =0withinintervalsforwhichJp <1andatothertimes, time for n ion zones, is Pˆ = nP , where P is the band
3 3 3
whenJp =J,representsthestorageofexcessprotonsinthe separation in the usual notation by which subpulse drift is
atmosphereatz>z .ThetotalcurrentdensityJ isfixedby described, and here is given by
1
theboundary conditions and at z =0 differslittle from the
P =τ +τ =(K+1)τ (37)
Goldreich-Julian value. Thus Jp = J and Jz = 0 until the 3 gap ee p
instantatwhichtheatmosphereisexhaustedandJp fallsto in the δ-function limit of the diffusion function fp. In this
someresidual valueJp <J.Then ion flowrecommencesal- system, the velocity u˙ is determined by n for fixed values
most immediately (in a time much shorter than thegrowth ofK and τp.Forthesame reason,drift direction isalsoun-
timeforspontaneouscurvatureradiation pairs) andcontin- specifiedandthereisnothingtoprecludeachangeinnora
ues until proton diffusion grows sufficiently to re-form the reversalfollowing somedisturbancetothepolar-capsurface
proton atmosphere. The durations of the time intervals for such as might be a consequence of the kind of mechanical
ion andproton emission arethoseforelectron-positron pair instability briefly described in Section 4. The drift time is
creationorotherwise,andlabelledτ andτ aregivenby, determinedprincipally bydiffusion through themoredense
ee gap
layersoftheatmosphere.Fromequations(13)and(14)itis,
τee
J = K dt′fp(τee−t′)Jz(t′) z1 dz l k T 1/2
KZ0 τee τp =Z0 v¯(z) =(cid:18)αgλAR(0)(cid:19)(cid:18) mBp (cid:19) . (38)
τ = dt′Jz(t′) (35)
gap J Z InSection3weobservedthatatmosphericdensityatz =0is
0
anexponentialfunctionofT >T sothatitsvalueisessen-
c
Therefore τ and τ both depend on the smallness of f tially unpredictable. Therefore, it is likely that the density
ee gap p
for small values of t−t′. In the δ-function limit for f , we discontinuityat z=0issmall andthatthereverse-electron
p
have τ = Kτ . Estimates of the parameter K = K Zν showersmaybecontainedentirelywithintheatmosphere.In
gap ee 0
were given in Paper I, but are repeated here, this case, the lower integration limit in equation (38) must
be replaced by z > 0. The diffusion time is then not di-
sm
K0 = 2.8hZsmi−0.76B102.62T6−1.0, B12 >1, rectly a function of B, but is dependent on surface gravity
K = 1.6hZ i−0.76T−1.0, B <1, (36) andonZ˜.Butitsdistributionforallpulsarsshouldbecom-
0 sm 6 12
pact.ThedistributioninthevaluesofK,givenbyequation
withν =0.85for10<Z <26.Intheseexpressions,T isnot (36),isprobablythemoreimportantsourceoftheobserved
thelocalsurfacetemperaturebutisanaverageforradiation spreadinthevaluesofP .Detailedcalculationofτ hasnot
3 p
emittedoverthewholepolar-cap,andB istheactualpolar- beenattemptedhere.Inparticular,ourestimatesofλ and
R
cap magnetic flux density. For B12 ≫ 1, the values of K0 lA are subject to some uncertainty.
given need to be modified, though not greatly, to allow for The fact that P is constant for a particular pulsar
3
thehigh-B values of Wp contained in Table 1. whereas the circulation time Pˆ3 is dependent on n could in
Guidedbytheestimatescontainedinequations(6)and principle allow comparison with E×B drift-velocity polar-
(7),weanticipatethatexceptforaverysmallnumberofneu- cap models in which Pˆ is constant. It is also worth consid-
3
tronstars,thesurfacetemperatureisatalltimesT >T so eringtheeffect ofmechanicalinstability inallowing leakage
c
that the case (ii) boundary condition is always maintained of protons(or low-Z ions) tolimited areas of thesurface as
and ion emission is never temperature-limited. It is limited described at the end of Section 4. The excess protons form
insteadbythefactthattheprotonatmosphereformsabove alocalizedatmosphereofgreaterdepththaninsurrounding
the ions and the protons are preferentially accelerated, as areas, so that ion zones (with consequent pair creation) do
described in Section 3. The only effect of the temperature not form there until it is exhausted. The observer of radio
variations described byequation (11) that occuras a result emission probablyseesplasmafrom nomorethanastripof
ofalternatingprotonandionemissionistochangetheden- polar-capareaatroughlyconstantrotationallatitudesoitis
sity of the LTE ion atmosphere. It is also worth observing possiblethatintervalsofnullemissionwouldbebeobserved
that the local nature of equation (34) is not disturbed by at fixedlongitude in a sequenceof subpulsebands.
the presence of E×B drift above the polar cap. Although In a survey of 187 pulsars, selected only by signal-to-
this slightly displaces a reverse electron shower relative to noise ratio, Weltevredeet al (2006) found that reversals do
10 P. B. Jones
occur and that roughly equal numbers of pulsars have sub- derivedfrom H isreachedat t≪τ .Thecooling at t>τ
0 p p
pulse drift to smaller or greater longitudes. They also com- isalso rapid so thatthemajor part of theX-rayluminosity
ment that subpulse drift is so common a phenomenon that is that of a black body at T and of area approximately
0
it cannot depend on pulsar parameters having extraordi- equal to the canonical dipole-field area 2π2R3/cP divided
naryvalues.ThebandseparationP isindependentofradio byK+1.
3
frequency, and is not correlated with period P, age P/2P˙, There have been many attempts to measure the polar-
or with the inferred dipole field B . Measured values of P cap blackbody temperature and source area of a sub-
d 3
givenfor77pulsarsinTable2ofWeltevredeetalaremostly set of radio pulsars. We refer to Zavlin & Pavlov (2004)
contained within a single order of magnitude, 1 < P < 10 for B0950+08 ; De Luca et al (2005) for B0656+14 and
3
s although the distribution has a small tail extending to B1055-52; Tepedelenlio˘glu &O¨gelman (2005) for B0628-28;
∼ 20 s. This is quite compact (for a neutron-star parame- Zhang, Sanwal & Pavlov (2005) for B0943+10; Kargaltsev,
ter) and is not inconsistent with the interpretation of sub- Pavlov & Garmire (2006) for B1133+16; Gil et al (2008)
pulse drift given here and with equation (37). The range of for B0834+06; Misanovic, Pavlov & Garmire (2008) for
P ,withestimatesofK foundfromequations(36),indicate B1929+10. Five pulsars (B0628-28, B0834+06, B0943+10,
3
τ ∼ 10−1 −100 s and N (0) = 1023 −1024 cm−3, values B1133+16, B1929+10) have source areas one or two orders
p A
that are byno means unreasonable. of magnitude smaller than the canonical area, but unfortu-
It is necessary to compare our polar-cap model with nately,a systematic comparison with equations (40)-(42) is
two other sets of pulsar observations. Given that protons not possible because in most instances theauthors are able
are the major fraction of particles accelerated and produce to say only that the observed X-ray spectrum is consistent
no electron-positron pairs, it is worthwhile considering the with the stated temperature and source area. The quoted
radio luminosity L . The order of magnitude of this can be source temperatures are ∼ 3×106 K and are larger than
ν
expressed as, those predicted by equation (42) unless it is assumed that
B ≫B . They are also uncomfortably large in the context
2π2R3B 1.4×1030B d
L ∆Ω∆ν ≈ dǫ≈ d12 ǫ (39) ofE×Bdrift-velocitypolar-cap modelssuchasthatdevel-
ν ceP2 (cid:18) P2 (cid:19)
oped by Gil, Melikidze & Geppert (2003) although it must
intermsoftheneutron-starradiusRandtheinferreddipole beconcededthatthismodelallows aninterestingtestofits
field B . From this we can estimate the energy ǫ radi- validity (see Gil, Melikidze & Zhang 2007). However, the
d
ated into solid angle ∆Ω within bandwidth ∆ν per unit case (iii) surface electric-field boundary condition on which
charge (baryonic or leptonic) accelerated at the polar cap. thesemodels relycould bemaintained only forion cohesive
Using the 400 MHz luminosities listed in the ATNF cata- energiesof∼10keVwhichinturnwouldimplyactualfields
logue (Manchester et al 2005) and a bandwidth ∆ν = 400 twoordersofmagnitudeormorelargerthantheATNFcat-
MHz, we find by evaluating equation (37) for a small sam- alogue dipole field.
ple(B0826-34,B0834+06,B0943+10,B0950+08,B1055-52,
B1133+16, B1929+10) that typical values are in the inter-
val ǫ=(40−400)δΩ MeV. Even though the emission solid
6 CONCLUSIONS
angle is likely to be as small as ∆Ω ∼ 10−2−10−1 sterad,
theremustbesomeconcernherebecausethenumberofICS Thispaperisacontinuationofapreviousstudy(PaperI)of
pairsproducedperprimaryacceleratedpositron(Harding& isolatedneutronstarswithpositivecorotationalchargeden-
Muslimov 2002) isnotlargesothateithertheconversion of sity and surface electric-field boundary condition E·B =
electron-positron energy to coherent radio emission is effi- 0 at the polar cap. The reverse flux of electrons arising
cient or other plasma components are involved, as in the from photo-ionization of accelerated ions is incident on the
paper byCheng & Ruderman(1980). neutron-star surface and produces protons through forma-
Asecondsetofobservationsthatarerelevantarethose tion and decay of the giant dipole resonance in the later
of polar-cap blackbody X-ray luminosities. The blackbody stages of electromagnetic shower development. Protons are
X-rayemission expectedfromapolar-capofionandproton themajor componentoftheaccelerated plasmabutwefind
zonescanbefoundfromequation(11).Thereverse-electron that a time-independent composition of ions, protons and
flux from photo-ionization heats an ion zone at a rate H0 positronsisusuallyunstable.Theconsequencesofthesephe-
within a time interval 0 < t < τp. An estimate of this can nomena should be observable in pulsars unless, of course,
beobtaineddirectlyfromthemeanelectronenergyperunit rendered nugatory by some factor not properly taken into
nuclear charge accelerated, given by equation (29) of paper account here. But we emphasize that there are almost cer-
I, and is, tainly other sources of instability contributing to the com-
H =6.0×1018Z0.85(0)B1.5T−1P−1, (40) plexbehaviourobservedinradiopulsarsthatmaybepresent
0 12 6
in each of cases (i) - (iii).
inunitsofergcm−2s−1.Inthisexpression,T isnotthelocal There are two instabilities which result in transitions
surface temperaturebutisan average for radiation emitted between states of different plasma composition abovelocal-
over the whole polar cap. The surface temperature derived izedareasonthepolarcapanditissuggestedherethatthese
from equation (11) with neglect of radiative loss is arethebasisforthecommonlyobservedphenomenaofnulls
T(t)= 2H0 t1/2, 0<t<τ , (41) andofsubpulsedrift.Thebasicunitsoftimeareτp ∼10−1
(πCλ )1/2 p s for the short time-scale instability and for the medium
k
time-scale, t ∼ 102 − 103 s. These instabilities are not
andincreasesveryrapidly,sothatthelimitingtemperature, rl
primarily electromagnetic in origin and will not be present
T ≈0.6Z0.17(0)B0.3P−0.2, (42) in neutron stars with negative corotational charge density
06 12