19 6 9ApJ. . .157.13950 2lz21 2 5052 15 24 3 -4 neutron starhavingadipolarmagneticfieldwhichisnotparalleltotherotationaxis.Wethenshowthat we predictthatfewpulsarsshouldbefoundwithperiodsinexcessof1.5seconds,alsogoodaccord and increasesintherotationperiods,thatemittedlow-frequencymagnetic-dipoleradiationis quadrupole radiation,thattheseenergylossesareinevitablyassociatedwithofangularmomentum The AstrophysicalJournal,Vol.157,September1969 in somedetailPaperIII.Forourpresentpurposeofdefiningthemodel,twosets with observations.Wedonotdiscusstheoriginofpulsesthemselves. celeration mechanismcanproduceprotonswithenergiesuptomc{e/Gm^)lor10eV,whichis accuracy) andtopredictthesofarunobservedsecondderivativeofperiod{dP/dP).Wealsode- extremely efficientatacceleratingchargedparticlestorelativisticenergies.Anexplicitexpressionfor such starsmaybeexpectedtoemitlargeamounts(10-10ergs)ofmagnetic-dipoleandgravitational- model. Wefirstmakeplausibleandthenassumethattheseatofpulsarphenomenonisarotating both numbersfoundtobeingoodagreementwithindependentobservations.Inextremecasestheac- The individualpulses,whichfillonlyasmallfraction(~1/20)ofthepulseperiod,are and theirclasscharacteristicsarebecomingapparent.Wewilldiscusstheobservations the mean,forpulsars.Finally,afterdeterminingmagneticdecaytimetobeabout4millionyears, tween period,P,andrateofchangedP/dt»3X10“/P(sec),whichiswellobserved,in somewhat inexcessofthemostenergeticcosmicraysyetobserved.Thetheorypredictsarelationbe- termine theluminosityofnebulaandhighest-energyelectronspresentlybeinginjectedintoit— the periodasafunctionoftimeallowsustocalculateageCrabNebula(with»20percent characteristics aresignificant.First,thepresentlydiscoveredpulsarshaveperiodsP emitting periodicpulsedradioradiation,atotalofabout40pulsarshasbeendiscovered, distribution ofpulsarsshowsthattheyaregalactic objectsofdiskorPopulationItype Maran andCameron1969);wewillreviewthe arguments verybriefly.Theangular (c) 1969TheUniversityofChicagoAllrightsreservedPrintedinUSA often stronglypolarized.Forthesecondsetofcharacteristics weexaminethedistribu- and ratesofchangeperioddP/dtsatisfyingtheinequalities tendency towardclusteringatthegalacticcenter, andthepresenceoftwopulsars available data(seePaperHI)showastrongconcentration inthegalacticplane,no tion ofpulsarsoverthesky.Althoughobviously influencedbyselectioneffects,the and and enablesustorestricttypicaldistancesroughly totherange10pcAstrophysics DataSystem We presentinthispapertheinitialinstallmentofaquantitativeexplorationoneparticularpulsar Since theoriginalannouncementbyHewishetal.(1968)ofastronomicalobjects The conclusionsdrawnfromthisdataare,bynow, wellknown(seeWoltjer1969;and VeryrecentlyReichleyand Downs(1969)andRadhakrishnanManchester appearto ON THENATUREOFPULSARS.I.THEORY J. P.OSTRIKERANDE.GUNN 13 0.0 0.3sec(FaulknerandGribbin1968;OstrikerTassoul1968),neu- stable, astronomicalobjectcanoscillateinitsfundamentalmode—whitedwarfshave jects intheusualelectromagneticspectrum. We wouldliketostressthatthepulsarscannot be axisymmetricstars,since,fromany work ofBaadeandZwicky(1934)OppenheimerVolkoff(1938).Others,includ- single class;ingeneral(period)oc(density)“,andanextremelywiderangeofdensity of whitedwarfsaretheoreticallypossible,buttheverywiderangeobservedperiod work withtickrateofabout1persecond?Stellarpulsationsanysingleclassstar tron starshaveperiodsP<0.05sec(MeitzerandThome1966).Overtoneoscillations are notalikelyexplanation.Theshorterperiodsintherangewhereanyknown, do emitpulses.Tothelowestorderinangular velocity,fí,theenergylossesfrom However, thefirstassociationbetweenpulsarsandsuchstarswasmadebyGold(1968, rotating (Hoyle,Narlikar,andWheeler1964;TsurutaCameron1966),thatthe stars wouldbeformedduringsupernovaexplosions,thattheyinitiallyrapidly mestic time—therotationofamassiveobject.White-dwarfstarswerefirstsuggested radiation ofenergywouldleadtoperiodchangesmuchgreaterthanobserved—and would berequired.Finally,theperiodsareverystable,withcharacteristictimesr= relativistic particlesbythisradiation.Conclusions,asummary,andbriefdiscussionof selves. SectionIIcontainsadiscussionofthemultipoleradiationwhichisnecessarily many implicationsofGold’ssuggestion.Wewillnottreattheoriginpulsesthem- polarization andincreaseofperiod.Intherestthispaperweshallexploresome of «10gauss,couldaccountformanythepropertiespulsarsincludingpulse ing Zwicky(1938)andColgateWhite(1966),hadsuggestedthattheseneutron a verymuchshortertimescale. the wrongsign. and are notsymmetricabouttherotationaxismustradiate energyandangularmomentum. applications arereservedfor§IV. emitted bynonaxisymmetricrotatingstars,and§IIIdiscussestheaccelerationof time-varying gravitationalandmagneticfieldsare (LandauandLifshitz1951) aspect, theappearanceofanaxisymmetricstaris independentoftime,andthepulsars energy sourceoftheCrabNebulamightbearotatingneutronstar(Wheeler1966). 1396 (Ostriker 1968),butevenintheircentralregions,whitedwarfscannotrotatewitha (dP/Pdt)* asgreat10years,whereastheproposedpulsatingstarswouldevolveon (over 100-fold)isdifficulttounderstandifpulsarsare,infact,pulsatingobjectsofany 1969), whoarguedconvincinglythatrotatingneutronstars,withsurfacemagneticfields krishnan etal,(1969),showsthat anoff-axismagneticfieldisindicatedforthisobject. 2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The clockcannotbeprovidedbyorbitalmotioninabinarysystem,sincegravitational There remainedthepossibilitythatpulsartimeiskeptbysamemeansasdo- What propertyoflow-luminositystarscanprovidesuchextraordinarystableclock- Any object,rotatinginvacuo,havinggravitational, electric,ormagneticfieldswhich Amoredetailedargument,based onthetimedependenceofpolarizationinPSR0833 byRadha- a) LossesofEnergyandAngularMomentum J. P.OSTRIKERANDE.GUNN dE^/dt =-\2mPQl/â (1) dEJdt =-^GDPW/â. (2) g II. MULTIPOLERADIATION Vol. 157 19 6 9ApJ. . .157.13950 1/2 12 112 30 3 28 2-37 612 6 be safelyassumedtoremaininahighlyconductingstateuntiltheultimatecollapseof where (a)is(§)timesthemeanradiusofgyration, andeistheellipticityinequa- be computed,asweshallsee,fromthéobservedpropertiesofpulsars. lar componentofthemagneticmoment, lar totherotationaxis(Preston1967),Withasurfacefieldof«10gaussperpendicu- radius 10cmandsurfacefield(£)ofgausswillevolveintoaneutronstar population, sothatmoderate-sizedfieldscanbeexpectedtooccurintheprogenitorsof is numericallyofordera{B)«10esu.Thisquantity,roughlyestimatedhere,can to bebestdescribablebyamagnetic-dipolemodelwiththefieldaxisnearlyperpendicu- the magneticandrotationaxeswillbealigned;indeed,Astarsseem fields areknownamongthe(relativelycommon)Apstars.Theinteriorsofsuchstarscan for starsslightlyyoungerandmoremassivethantheSun,thoughofcoursemuchhigher of theellipticitylater(AppendixA);atthistime weshallonlyinvestigateitsconse- torial plane.Themeanradius(a)isverynearlythe physicalradiusaforourstar:1.1X the core,sofieldwillevolveinsuchamannerastoconserveflux.Thusstarof vector mi,themass-quadrupoletensorDab(fromwhichmiandDiarederived), ments perpendiculartotherotationaxes.Thereisnoelectricdipoleradiation(bysym- neutron-star modelofHartleandThome(1968)constructedwiththeVequation moment-of-inertia tensorIa?ThesequantitiesandtheobservedangularvelocityQwill nected witharotatingneutronstar,howdoweestimatethemagnetic-dipole-moment and parttorelativisticeffects(HartleThorne 1968). Weshalldiscusspossibleorigins aspherical bodyisgivenby almost rigidly(i.e.,onewhose figurechangesverylittleinonerotation)and losing energy also someevidence(cf.PaperIII)thatpulsarsareassociatedwithayoung,massive dence (see,e.g.,Wilson1963)thatyoungstarshavelargerfieldsthanoldones;thereis state, thentheradiusaandmomentofinertia/(=fS/u)are metry), andtheelectricquadrupolecanbeshowntolessthangravitational Here andmiarethecomponentsofmass-quadrupolemagnetic-dipolemo- No. 3,1969 quences. the neutronstar.Fieldsoforder100gaussseemnotunreasonable,asanaverage, when rotationaldistortionisnotsignificant(fí<Vß288i /ß, (5) e mi —msina(cos=m*fí/wí2),(4) PULSARS 1397 19 6 9ApJ. . .157.13950 24 2 4 by anycombinationofmultipoleradiation,mustloseangularmomentumJalsoatthe rate since HartleandThorne(1968)havecalculatedthelowest-orderrotationalcorrection inertia veryclosetothoseofnonrotatingneutronstars.WewilltakeIbeconstantin and, asaconsequence,itsrotationperiodmustincreaseatpreciselycalculablerate. most rapidlyrotatingknownpulsar(NP0532)£2<4X10sec“,wemaytreatpulsars still easilyintegrable,buttheadditionalcomplexityofsolutionmasksitscharacter the presentwork.Onecanofcourseincludedistortingeffectsrotationifonewishes as slowlyrotatingstarshavingnearlysphericalequipotentialsurfacesandmomentsof Two limitingformsofequation(10)areinterest. Thegravitational-quadrupoleradia- that aisconstant),returningtoexaminethisissuein§lid.Underthesecircumstances and addsnothingofphysicalinterest.Wewillalsoassumethatthereisnoalignment(i.e., to themomentofinertia(/=/+const,fi);resultingequationsmotionare the totalenergylossappearsasaofrotationalkineticT.Fromequations(1) We havetakenouroriginoftimetobeatthefiducialinstantwhenfí=.Theinitial Equation (9)canbeintegratedtogivethenormalizedperiodximplicitlyintermsof Now let the pastduetoitsstrongerdependenceonangular velocity.Duringthisera and (2)wehave 1398 J.P.OSTRIKERANDE.GUNN However, afteratime tion willalwaysdominateoverthemagnetic-dipole radiationforanintervaloftimein time, b,whentheperiodwasveryshort,isgiven by where fioistheangularvelocityatsomefiducialepoch.Equation(8)becomes time t: 0 0 28 4 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem For aneutronstar,rapidrotationcorrespondstoO«10sec“.Sinceforeventhe Thenotationusedherediffers fromthatinGunnandOstriker(19696). K2 } dt W¿/md Tm — 2T 2otÆ ’°-GDJW 0 3 3c/ b) IntegrationoftheEquationsMotionStar 2 t =èr[x-1bn(^y)]•(10) m /i =-|r[l-bn(1+„)]. (11) m t =—b+IrgX*,rçr*<$C1. (12) dt A 5 45c/ W Tt ~ + (13) dJ _1dE 24 J (dE\ _2mxQ__DJG \dt / = * “fíPo _ OoP gq 3 3 c and 5 45c 6 Û. T m Vol. 157 (8) (7) (9) 19 6 9ApJ. . .157.13950 2 49 12 4-1 4 -1 38 38-1 Let usnowdefineacharacteristictimeto: pletely determinedandwecancalculate,amongotherthings,ti.Underthereasonable magnetic-dipole radiationdominates,andequation(10)approachestheform this isadirectlyobservablequantityifwetakethefiducialinstanttobepresent. No. 3,1969 If, inadditiontoto,weknowthevalueofparameterrç,thensolutioniscom- With theaidofequations(8)and(9)wecanrelatetovariousdecaytimes assumption thattheneutronstarwasformedrotatingrapidly,wearethusabletodate be pointedouthere,however,thatthisluminosityismanyordersofmagnitudelarger intense 30-Hzelectromagneticwaves,thefateofwhichweshalldiscusslater.Itshould it everwasburningnuclearfuel.Theradiation,ofcourse,isintheformenormously where %iistheinitialvalueofæ(#*«2X10~). the explosiveeventwhichproducedpulsar.Wefind,fromequations(11)and(16), dicted ageis1170yr,inerrorbyabout25percent fromtheknownageof916yr.Of present isessentiallytheage,—U,towithinafewyears. Ifthereisnogravitationalradia- and gravitational-quadrupoleradiationare Integrated overthelifetimeofobject,totalenergyoutputsinmagnetic-dipole magnetic timetisafewthousandyears(cf.eq.[8]);nearthebeginning,however, fluxes (Friedman1969).ThekineticenergyTis2.5X10ergs,sothecharacteristic than thepulseluminosity,includingopticalandrecentlydiscoveredX-raypulsed For theCrabpulsar,ß«200,and,withasurfacefieldofabout10gauss,(dE/dt)d time scalewasmuchshorter.Atß=10sec,themagneticisafewyears,sothat parameter 77isdetermined(77«5)theellipticity is«3X10~(cf.AppendixA),and, radiation hashadasignificanteffectontheevolution oftheCrabpulsar.Ifwerequire Paper II,themostplausible(oratleast interesting)isthatgravitational various possibleexplanationsforthis“discrepancy,” whichisdiscussedfurtherin the Crabobject,to=2340yr.Thus,inabsence ofgravitationalradiation,thepre- tion (I/77=0),theageisjustto/2.Iftherenomagnetic radiation,theageisto/4.For the timeintervalbetweenanepochatwhich objectwasrapidlyrotatingandthe gravitational radiationtobeofsuchamountas removetheagediscrepancy,then and (18) sec) isintheformof^60-Hz gravitational-quadrupoleradiation. at thepresenttime,one-sixthofluminosity oftheCrabobject(Lq«10ergs radiation nowisanappreciable fractionoftheenergyoutput,itmusthave dominated —10 ergssec.Thusitseemsquitepossiblethatthestarismoreluminousnowthan w m ? g © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Let usconsiderbrieflytheorderofmagnitudequantitiesweareinterestedin. Consider nowtheearlyhistory ofapulsarsuchastheCrabobject.Ifgravitational age =-U¿to(l+1)[l-iln(1„)].(17) t = 2 2 AE =—J/ßo??In(1+lAiA), 1 md A£ =-ilß[*r-9?In(1+1AA)], gq0 T =(1/T+I/t^)“t(11A)".(16) 0mw To = J 1 SldtJo ^\Pdt) Q i day,/idpy. = PULSARS 5 rçr »1. 1399 (14) (15) 19 6 9ApJ. . .157.13950 53 4 1238 371 -1 12 The initialluminosityinmagnetic-dipoleandgravitational-quadrupoleradiationare in thepast,when£2waslarger.Ifinitialangularvelocity£2¿isassumedtohavebeen percent, andthat,ifallowanceismadeforgravitationalaswellmagneticradiation, form ofgravitationalradiation.Duringthelifetimeobject The initialkineticenergyisabout1.5X10ergs,mostofwhichis,course,lostinthe They becomeequalattime about 10,thentheinitialmagneticandgravitationaltimesare magnetic moments.Expressingtheresultintermsofmeansurfacefield(B),and energy sourceforthesupernovaeventitself.Weshallreturntothispointlater. roughly halfofwhichisemittedinthefirstyear.Thustimes,energies,andluminosi- short-period pulsarsisone ofage,andso,barringfielddecayorsecular-orientation ef- ing tofieldsofafewhundredgaussonthemain sequence, areinvolved. of theCrabNebula.Theoutput,withexpected field(10gauss)assumed,is10 in theformofultrarelativisticelectrons(Haymes etal.1968)tosustaintheluminosity of theassumptionanymechanism)andestimate of^7X10ergssec“required around, and,onthebasisofobservedvaluestoinvarioussources,compute to supemovae,andindeedtherotationalenergyofremnantmaywellbeprimary ties, inusable(i.e.,electromagnetic)formsarecomparabletothequantitiesappropriate assuming thatsina=1wehave,fromequation(8), the discrepancycanberemovedentirely.Wearenowinapositiontoturnargument angular momentumareviamagnetic-dipoleradiation, thenreasonablefields,correspond- called attentiontothecloseagreementofthisnumber (whichis,ofcourse,independent Or, ifweneglectthegravitational-radiationcorrectionsinequation(16),(19) the following:Iftheseideas arecorrect,theprimarydifferencebetweenlong-period and ergs sec,andalsoinfairagreement,whichisto saythat,ifthelossofenergyand For theCrabobject,{B)is2.6X10gaussiftherenogravitationalradiation,and can berewritten in ourcaseis about 8percentsmallerifthereis.Thetotalluminosity,ofcourse,isjustISldti/dt,which 1400 J.P.OSTRIKERANDE.GUNNVol.157 8 8 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The authors(GunnandOstriker1969a)others (cf.FinziandWolf1969)have We havealreadymentionedthattheageofCrabispredictedcorrectto—25 A muchmorepowerfulcheck onthereasonablenessofmagnetic-dipole pictureis 45148 Lom —5X10ergssec“andLo^3. g 2 Tmi =TmXi«0.89yr,TgiTgXi*0.016. A2 4x <> -Gâÿ?)“•0») c) ObservationalChecksoftheTheory 381 Ltot =7.1X10ergssec“. (21) 51 AE «—5X10ergs, md te —ti«80yr. (JP\U2 P1/7? =0(20) Wo ’- 19 6 9ApJ. . .157.13950 2 26 The mostimportanteffectofadecayingfieldisthattheperiodwillnotbecomein- face integral,enclosingtheobject,offield-stress energytensorWefind arrow isCP0808,whichhasonlyanupperlimitondP/dt. where were tohappen,a—>0andthemagnetic-dipole radiationwouldbeturnedoff.The in thissectionbutwhicharederivedontheassumptionthatm—exp(—t/td). torque Nonanobjectduetoelectromagneticstresses canbeexpressedintermsofasur- definitely longbutwillstopchangingafteritreaches amaximum are stillofinterest.InAppendixCwepresenttheformulaewhichanalogoustothose The results,plottedinFigure1,showourexpectationsstrikinglyconfirmed;mostofthe pulsars asamongtheshort-periodones.Period-changedataoneleven(Maran fects, weshouldfindthesamedistributionoffieldstrengthsamonglong-period Canuto (1969).ThetimesarelongcomparedwiththeageofCrabsource,butthey magnetic-dipole fieldmaydecay.Estimatesofthemagnetic-decaytimetd~4aa-/ No. 3,1969 and Cameron1969)havebeenusedtoderivethesurfacefieldsbyusingequation(20). object. theoretically derivedsurfacefieldsarewithinafactor2ofthevaluefoundforCrab (ttc) «4X10yrcanbemadewiththeaidofrecentconductivitycalculationsby © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Is thereanytendencyforthemagneticandrotation axestobecomealigned?Ifthis In thisdiscussionwehaveneglectedthepossibilitythat,overcourseoftime, Fig. 1.—Derivedsurfacefields(perpendiculardipolecomponent)plottedagainstperiod.Pointwith 1 T =g-\(—BB+±B% ) +(—EE±E%].(23) bcbc ICbcC 1/2 Pmax =Po(l+td/T)• 0 abc Nd =€abJ'%Td'2, (22) dc d) Alignment PULSARS 1401 19 6 9ApJ. . .157.13950 7 23 particle releasedatthesurface ofthestar.Themotionto“wavezone” (fo=c/O) duced, underthephysicalconditionslikelytobepresent neartheneutronstar.Considera is verycomplicatedanddepends onthefieldsinnear-andintermediate zones,which will bestronglyaffected bythecorotatingmagnetosphericplasmawhich Goldreich tion produced,andtotheequallyimportantquestion ofwhetheritinfactcanbepro- for atleastthefirst10years. dipole. Inanycase,theobservationsindicatestrongly thatalignmentdoesnotoccur field, andthiscomponentcanbeexpectedtolarge, possiblyofthesameorderas of therotatingpointdipole,onecalculatesNtobeunchanged,~Nsomewhat field nearthestar.Ifoneconsiderstobeapproximatedbythatofarigidly these effectscanbedominatedbyanypermanentquadrupole componentofthemagnetic a timescaleofapproximatelyr(DavisandGoldstein1969).Itisclear,however,that altered, andiV*tobe0(+wí2íTsin2a).Thisfieldconfigurationpredictsalignmenton rotating, magnetized,andperfectlyconductingsphereinvacuo(Deutsch1955),instead The firstofequations(27)issimplytheformequation(10)takeswhen(l/rç—^O), alignment isextremelydifficult,dependingasitdoesondetailsoftheelectromagnetic or about1dayfortheCrabpulsar.Fromatheoreticalpointofview,question second showsthatthereisnotendencyforapointdipoletoalign,andthethirdindicates that therotationaxiswillprocessaboutmagneticwithaperiodP=(27r/i2) The componentNis,asweshallsee,theoneresponsibleforslowingrotationof Integrating equations(26)byusing(25),wefind star; itcanbedeterminedverysimplyfromaconsiderationoftheradiationfieldalone which thetorqueshavebeencalculated.Forasphericalstartheyare The componentsofthetorque,calculatedforasphericalstarfromequations(22), zV [Gunn andOstriker(1969a)]. m and (24),are The fieldofapointdipole,expressedinbody-centeredsphericalcoordinateswithpolar vp z axis indirectionQ,is(E=0) 1402 r © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We nowturntothequestionoffatelow-frequency electromagneticradia- Euler’s equationsforthemotionofarigidbodycanbetransformedtoframein Be =msm Br = 112 0 =Oo(l+U/rm)-,aconst.QÜ.(27) p dü/dt —N/I,Üda/dt=—N(Qdÿ/dt)sina/I.(26) = msm zxy iV* =0, 2m sma-rcosó-\—r-0Hr-6, in a-C0Ssincos024 í ~1^)*§~c+^“-() ¡„.[-|C0S* +(i-5)¡n*]. ¿S 3 La a?c.1a r 1Q.2m Ny J. P.OSTRIKERANDE.GUNN III. PARTICLEACCELERATION 2 2 2mQ Zac d) Introduction cos asin, at \aTm*to/ N z 23 2m£l 3 Zc 2 sin a . Vol. 157 (23), (25) 19 6 9ApJ. . .157.13950 li2 5 bundle, eventhoughmaterialisstreamingoutandnostationaryequilibriumobtained. Thus weexpectplasmaofnumberdensitytheorderthatgivenbyequation(30) is theradiusofwavezone.Theplasmainthisregioncandiffuseintozone The areaofthisbundlefieldlinesis netic Astars. plasma field,which,foroff-axisdipoles,isforbiddinglydifficult.Wesubstitute,the approaching thevelocityoflight,chargedensitydecreasesatleastasrapidly field, however,onecaneasilyshowthattheselinesoriginateinanareanearthepoleof Thus, present, averyroughqualitativeargumentofthenearfieldzone,followingspirit below beginstooperatewellwithintheradiusr. at r.Themotiontherewillalmostcertainlyberelativistic,sincethemechanismoutlined angular extent reach intothewavezone(r>c/Q).Whichfieldlinesthesearedepends,ofcourse,on wave inamediumwithinwhichthefrequencyispossibly muchsmallerthantheformalplasma uqj andtravelingtor,acquiresanenergy Ostriker 1969Ô).Aparticleofmassm,startingat restatrwherethegyrofrequencyis tion, theycanbeintegratedinasimpleand straightforward manner(Gunnand and thenweshallconsiderthereactionofanassembly oftestparticlesonthefields. Now thatplasmawhichultimatelyescapesdoessoinitiallyalongthosefieldlines formal gyrofrequency,equation(29)becomes the treatmentbyJulianandGoldreich(1969). tion, butareallysatisfactorysolutionawaitstreatmentoftheproblemcoupled frequency. However,thisstandard criterionisirrelevantinthepresentcase(thestrongfield limit)where Some ofthesepointswerefirstconsideredbyDeutsch (1955)inconnectionwithmag- If welet=(Awep/ni)betheformalplasmafrequency,andu—eB/mc the wave-particleinteraction is highlynonlinearandtheparticleenergyextremelyrelativistic Weat- (it ispropelledtherebytheinducedelectricfields),and,evenifitmoveswithavelocity the surfaceofstar,atleastinthoseregionswhereanequilibriumcanbeestablished. through thenearfieldiscurrentlybeinginvestigatedforoneparticularconfigura- the detailsofnearfield.Iffieldisqualitativelysimilartovacuumdipole No. 3,1969 tempt toderiveamorerelevant criterioninsectionIIIc. 1/A Then,ifequation(30)holdsatthesurface,itwillholdeverywherealong and thechargedensityis (1969) hasarguedmustsurroundarotatingmagneticstar.Theproblemofacceleration 0 0 c 0 ce 5 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Let usnowinvestigatethemotionofatestparticle onceithasreachedthewavezone, The expecteddensityoftheplasmamustbesuchastoresultinaforce-freefieldnear Noteaddedinproof:Wehavebeenreproachedforconsidering thepropagationofanelectromagnetic If weneglecttheCoriolisandcentrifugaltermsin thetest-particleequationsofmo- b) IntegrationoftheEquationsMotionaTestParticle 3 A «7rr/fo,wherer=c/Q(32) 0 2 |£| =_lí,XR|~tiB/r,(28) s Pc = 12 dp «(aß/c)'.(31) Up «Oco.(30) _1_ 47T PULSARS 3 47Tf * SlBs 1403 (29) 19 6 9ApJ. . .157.13950 ß Here Here <¿>oistheinitialphaseofparticlemeasuredwithrespecttocrest where well asthefieldconfigurationpresentedbyDeutsch(1955)forarigid,rotating,magnetic is thedimensionlesselectromagneticfieldtensor,andTareChristoffelsymbolsof we obtaintheequationsofmotion, Our finalformula(eq.[58])canbeobtainedbysubstituting(1—r/r)forIn(fAo)in about thepolaraxis(íí/fí): the analysisisconsiderablymorecomplicated,conclusionsareessentiallysame. Now One obtainsthisformfromconsiderationofthewavefieldrotatingpointdipoleas Here ristheparticlepropertime.Ifwenowdefinedimensionlessfourvelocity can nowbewritten Equations (38)canbesimplifiedbytheintroduction ofthephysicalvelocitycomponents the secondkind.Weconsiderfirstmotionofparticlesnearequatorialplaneforwhich electromagnetic wave.Wewillnowproceedwiththefullsetofequations.Although the magnetic-dipoleradiationisaplane-polarizedsphericalwave: For motionclosetotheequatorialplane(sin0^1, cos0~0),theequationsofmotion conductor. Wethenhave the aboveexpression. 1404 ta 0c © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We adopt,forconvenience,asetofdimensionlesssphericalcoordinatesmeasured 921 dw _(m)' /(m - 3 sm fl 2l3 ^ p p 123 Br =Er0,£*£,,=(B),BeEt0.(37) dv°/d^ =vf(iJL—p),dv/d%p[(v)+sin0(z;]f(p sP p1 r° ==ßQt,r=pril/c^6jc/>£Í2r.(33) u =V, 2s1 lz23 3 2 ^jz -vh—sin6cos(v)=(v°v,(38) —jz ~\—v+2^?;cot0=0, dv 2 dí p dÇ p d2 u =pv £=wc2ins J. P.OSTRIKERANDE.GUNN [?72Ci)^r- ßea all2 f{y) dv/d£ +Tv=f.(35) u =Igv(nosum) €aa aa = ßlßa fa =(c£i)~F(36) em z u* =psm _ (B)psesin(y) s vß EEEdrß/dii,(34) p) (pui) =(T Tt P and ^)/(m -P),(42) Vol. 157 (40) (39) (41) 19 6 9ApJ. . .157.13950 921/312 -36 pd 921/3 1193 particle’s velocityislargecomparedwithitsinitial velocitywhenwestartthispartof The firstofequations(43)saysthatthe^-velocity dropsas1/p,andweassumethatthe ly closetothevelocityoflightthatintegralexpression forthephasechange is highlyrelativistic,andforinitialvelocitieslessthanorofthisorder,relation(48) to 7,thesystemofequationsbecomes the integration,soweshallneglect0-motion.If nowchangetheindependentvariable change appreciably.Physically,thismeansthatthe particle’sradialvelocityissufficient- another assumptionwhichwewilljustifyaposteriori—that thephase(p—p)doesnot in theregionwhere7^>>aobutrelation(48) isstillvalid.Inthisanalysiswemake holds. good solongasu<(44) d e29 C - f(c-dr/dt)dt=/•(!-dp/dp)dß«1. (50) d£ ppo da _(u)(u*)Jo hMo y9 ¿(p«*)=0, g=e).(43) p2 e = (/o/po)êandui(/o/po)£.(47) du*/dt =0,dy/dÇ-(fo/po)u.(45) 9p 9 9 9 1 p du _a—u/fo yu/fda_u dy tt u dyfo* 0 a =y—uconstant.(46) 0 p p02 u =; u =|a(w) 0 PULSARS 1405 (51) (49) 19 6 9ApJ. . .157.13950 02 01 _1 Thus The integrationconstanthasbeensettozerosincey=(w)/2ainitially.Then which integratestogive The ratio(ao//o)issmall byassumption,sothefirsttermisnegligible; secondterm is justthecumulativephase driftarisingfromthefactthatterminalvelocity ofthe Now equations(53)and(55)give which yields 1406 where Thus, as{u,u’')gotozero,adoesnotapproachbutthesmallconstantvalue The equationofradialvelocityis and thiscanbeintegratedtogive just anapproximationtoexactintegralwhichcanbeeasilyobtained,namely,that particle. Here yisofcoursejustE/mâ.Thiscompletesoursolutionforthemotionatest v^Vfj =1.Ifweinsertthedefinitionofa,thisidentitybecomes and so o (27max)- Thecumulativephasechangeisfoundbyevaluatingtheintegral © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem How goodistheassumptionofconstantphase?Firstwenotethatequation(53) 2eI/13 a =(po/p),andu(2a)o/p)[3/o(|al—po/p)]- 0 J. P.OSTRIKERANDE.GUNN p 2 e (2)i/2^ dy (dy/dZ)(dy/dit)fou°’ dp u c3 m u=7 1/23 2ay =(m«)+(O1a.(60) ~ {-'°' 7 =[3/o(§a)(l-Po/P)?, 0 1/2 1/: da/dy =—/o“(2a7), a =^ao 9 9 du _g+yda/dy dy u œ dr 92 (u) =2ay. 1/28/2 ao --if7). 2/3/3 /2Y u’W =f^dp'JrId' P + p V2 \9f^) ’ 3/o V2 3/o ,3/2^ _ U PQ PO' yp Vol. 157 (54) (55) (53) (52) (61) (56) (58) (57) (59) 19 6 9ApJ. . .157.13950 e 7 1018 The sameconditioninsures thatthewaveequationforBisnearly E;thefields and theconditionforconsistencyisthatextreme righthandsidebesmall,or is oftheorderquantityitself),andweobtain term ontheleftdominatesfirst(assumingthat thechangewithphaseofanyquantity Now supposethat/decayswitheffectivelength (p),and(p))$>Po-Thenthesecond velocity, so The firstofequations(64)becomes remain equalandcrossed. Notethattheconditionbecomesmoreand favorable The current/,however,isessentiallyenjiP/7,whereuand7refertotheelectron0- Let E—mecfü/pe,asbefore,andchangeindependentvariablestoradiuspphase particles. Ifwetakeapointofobservationinthefarwavezone{rr), part ofthewavecarryingangularmomentum,aswellresidualdipolefield,purely the fieldequationscanbewritten the amplitudeofwave,/,nowdecaysasenergyistransferredfromwaveto behavior occurs. Trivelpiece 1968)didnotconsiderthelimitofveryhighvaluesof/=w/fíinwhichthis both effects,however,giveresultsnearlyidenticalwithourapproximateanalyticsolu- for thesakeoftractability.PreliminarynumericalcalculationsbyBraly(1969)including the particleviainducedcurrents.Theequationsofmotionareasbefore,exceptthat long beforethis.Wesee,then,thatelectromagneticwavesoflargeamplitudeforman tions. particle isslightlylessthanc.FortheCrabobjecttaking/=5X10(validforpro- or about2pc.Onecaneasilyshowthatinteractionwiththewaveisalreadynegligible tion; previousinvestigationsofnonlinearwave-particleinteractions(see,e.g.,Joryand tons), wefindthatthephasedriftisoforderunityforp~4X10,r«6cm, extremely efficientacceleratorofchargedparticles.Inthesecalculationsweneglectedthe No. 3,1969 0 0 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The primaryreasonthemechanismworksisconstant-phasecharacterofmo- If onehasanassemblyofparticles,mustinvestigatethecouplingwaveand 2 2 p dp£2’fí* 1 d(pE)E^4xdjpB)B47t. ( 2 2 2 2 E =Et,BB*,jje-,(63) c) PlasmaEffectsintheWaveZone p dpdpd\pd\p\Q7/ p dpdpdxp 1 dy_2d/_d_/V^\ 1 dy_2d/ = 2e112 (p> cop <2o£ly/u«2o£ly. (69) 1 xp =p—.(65) / dp ii/ PULSARS 2 Mc£t dip* e 2 ¿ 4:T6 dj 2ftt 7 pw u° p 1407 (68) (67) (66) 19 6 9ApJ. . .157.13950 319-1 2 2 12 2 1 of theCrabobjectaccordingtothismodel,showing themagnetic-dipoleandgravita- The theorygivesaquantitativedescriptionofthe motionofthestarandbehavior radiation ifmotionsinthewavezonearerelativistic.Ifonedropsfirstterm,equation tion oftime. tional-quadrupole luminosities, theperiod,andmaximumprotonenergies asafunc- capable ofshortingouttheassumedradiationfields. InFigure2wepresentthehistory tion occurs,thentheplasmainwavezoneis relativistic,andthatplasmaisnot of acceleratedparticles.Thesolutionisconsistent inthat,ifthemagnetic-dipoleradia- electromagnetic radiationwasshowntobeanefficient acceleratorofchargedparticles. orders ofmagnitudegreaterthantheobservedpulsed luminosity.Thelow-frequency of aboutafactor2,andtheluminositiesarein therange10-10ergssec,many large quantitiesofmagnetic-dipoleradiationverylowfrequency,themorerapidly luminosity. Thatconditioncanbewritten decay ofthewavefieldindicatedbyequation(71)willnotaffectsolutionsgiven instead ofthefamiliarœ)forparticleaccelerationsolongasthedecaydoesnotoccur(p/p~1). equation. Thephysicalreasonthatthepropagationconditionisgivenbyequation(69) where coisthegyrofrequencyofionsatwavezone(r=r). (67) canbeintegrated,andweobtain,byusingequation(43), Inserting physicalvariablesintoequation(70)andintegratingit,wefind 1408 p py t 0 î0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We haveseenthattheobservationsstronglysuggestpulsartimeiskeptbyrotat- IV. CONCLUSIONANDSUMMARYOFRESULTS 03sr(73) >2(B)aeZi’°^ s J. P.OSTRIKERANDE.GUNN 5 3C*Mi_i>7'2\ 234 47rro^^c7 (72) max 2 2 — =const.neMeCy,(71) 47T E e 2 cop dy dp * Vol. 157 (70) 19 6 9ApJ. . .157.13950 131/3 and electromagneticluminosities,respectively,Emaxisthemaximumprotonenergyproducedby acceleration mechanism. written (cf.eq.[58]) where AisthemassinatomicunitsandZ is thechargeinelectroncharges.For for ao^1(verysmallinjectionenergies),thisenergy issufficienttogeneratethemaxi- tion, canbeobtainedwithnodiscrepancy. with a20percenterror,andphysicallyreasonableamountsofgravitationalradia- mum observedfrequenciesinthesynchrotronspectrum oftheCrab.Itisnoteworthy NP 0532,equation(74)predictstheelectronenergy Eax=2XlQaoeV,andeven derived fromP,dP/dtandtheparticle-accelerationtheory—agreescloselywith observations. electron synchrotronluminosityoftheCrabNebula. that neithersignificantlylowernorhigher cutoffenergiesarepermittedbythe No. 3,1969 m y © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 2.—EvolutionoftheCrabpulsaraccordingtotheory.LqandLmaregravitational 3. Themaximumparticleenergiesattainablebythis accelerationmechanismcanbe 2. TheageoftheCrabNebulaiscalculated,inabsencegravitationalradiation, 4. Thederivedfieldstrengths forthosepulsarswhichhaveameasuredrate ofperiod In addition,severalresultsemergewhichcanbecompareddirectlywithobservations: 1. ThecalculatedluminosityoftheCrabpulsar(NP0532)inrelativisticelectrons— 141/32/31/3122 4/3 Pmax =1.30X10^4Zao(Ä/10)ß/200) eV,(74) PULSARS 1409 19 6 9ApJ. . .157.13950 19 6 49(638)-1555, andtheNational AeronauticsandSpaceAdministrationgrant NGL-31- is 1.38and1.12sec,respectively.Pulsarswillbeabletoreachperiodslongerthanthis 001-007. present, fromthevantageofthistheory,adetailed analysisoftheaccumulatedpulsar This sourceismorethansufficienttoaccountfortheentireoutputofaTypeIIsuper- The observedspectrumandfluxofhigh-energycosmicrayscanbeobtained(cf.Gunn observations. nova, andcontinuestooperateaftertheexplosion,thusprovidingexcitationenergyfor and Ostriker1969b)providedthathigherfieldsexistinsomepulsarsthantheCrab neutron starcausestheinitialejectionofsupernova envelope.Inthatpaperwewill mum protonenergythatcanbeproducedbythisprocessis(cf.GunnandOstriker19696) lived objectscapableofacceleratingprotonstoenergiesinexcess10eV.Themaxi- produced bythecorecollapseofsuchstarswouldbeextremelyenergetic,veryshort- peculiar objectswhichhavesurfacefieldsverymuchlargerthan“normal”stars.Pulsars derivative oftheperiodCrabsourceshouldbe equation (C9),weseethatthemaximumperiodCrabandVelasourcescanattain supports thecorrectnessof(P,dP/d/)-relationinourmodel(cf.eq.[20]): grounds seemthemostlikelyprogenitorsforpulsars,thereisasignificantfractionof topic willbedevelopedfurtherinPaperII,butafewpointsareworthmentioninghere. only iftheyhaveinitialfieldssignificantlygreaterthanthe“standard”sources. on thebasisofthistheory.Ifmagnetic-decaytime,td,is4X10years,then,using increase showasmalldispersionandnonoticeablecorrelationwithperiod(Fig.1).This cosmic rays,andtosomeaspectsofthepulsarphenomenon. PaperIIIinthisserieswill also presentapplicationsofthetheorydeveloped heretotheCrabNebula,galactic decays onatimescaleofabout5days—notexponentially,butonlyinverselyastime. in partbytheAirForce OfficeofScientificResearch,U.S.AirForce, contract AF object. Itisknown,however,thatamongtheearlyAandlateBstars,whichonother the secondresultobtainingifgravitationalradiationisimportant,firstitnot. greatly toourunderstandingoftheproblem nearfield.Thisworkwassupported seems possiblethattheaccelerationmechanismoutlinedinthispaperandappliedto large-scale magneticfields,hasbeenobservedinmanyastrophysicalcontexts.Itthus this problem,andtoProfessorPeterGoldreichwho, onavisittoPrinceton,contributed the expandingshelltoremainathightemperatures. independent ofthemassstarandcomfortablyinexcessanyobservedcosmic-ray energies. 1410 J.P.OSTRIKERANDE.GUNNVol.157 Crab Nebulamayhavemorewidespreadutility. 12 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We seethatpulsarsarecapableofproducingcosmicraysveryhighenergy.This The electromagneticluminosityofapulsarisinitiallyabout10Lo(cf.Fig.2),and One predictionemergesfromouranalysisofthemotionstar(§lib).Thesecond 5. Thefactthatfewlong-periodpulsarshavebeenfoundisquiteeasytounderstand We areindebtedtoProfessorRussellM.Kulsrud for manyprovocativediscussionsof In PaperIIwewillexplorethepossibilitythatinteraction withthefieldofrotating Nonthermal emission,presumablyassociatedwithrelativisticparticlesgyratingin 2-11 dP/dt =—0.016nsdayyror—0.021day“yr“, 221 £ =Wpc(-¿r)-10eV,(76) max 15 dP/dt «3X10“/P(sec).(75) 19 6 9ApJ. . .157.13950 2 6 pressure anisotropyhasthesamesymmetryasmagnetic field,wecanneglect,assecond- order terms,thequantities(M22—Mzz),(P22P33), and(In—/2)ß.Now,straightforward stresses, andtheanisotropicpartofpressuretensor. Thus,undertheassumptionthat algebraic manipulationofequations(AS)yields We areinterestedinthefirst-orderdeviationsfromsphericity producedbyrotation,magnetic Here Uiisthefluidvelocity.WehavetriviallygeneralizedChandrasekhar’streatmenttoinclude is easilygeneralizedtoarbitraryorientations.Forauniformlyrotatingstarwithangular and rotationalaxesofsymmetryareorthogonaldesignatedbyindices13;theargument an anisotropicpressuretensorpij.Letusnowconsiderthespecialcaseinwhichmagnetic velocity 0,thenontrivialvirialequationsare where tions aresignificant.Chandrasekhar (1969)hasalsoderivedthepost-Newtoniananalogues ofequations provide nineexactintegralconditionsontheequilibrium.Theseequations,derivedandex- to ^20percentinderivedquantities maybeinvolved. tensively developedbyChandrasekhar(1961),canbewritten lij isdiagonal.Thusourinquiryintotheequilibriumstructurereducedtoadeterminationof The mass-quadrupoletensorissimply three numbers—thediagonalpartsof/*,•.Wenaturallylooktothevirialtensorequationswhich It iswellknownthat,ifwechooseascoordinatedirectionstheprincipalaxesofbody,then of thedensitypdesignatedbymoment-of-inertiatensor star, wedonotneedtoknowtheequilibriumstructureindetail.Werequireonlymoments (A3), butwewillcontinuewith thesimplerclassicalsetofequations,bearinginmindthat errorsofup 2 EQUILIBRIUM SHAPEOFAROTATING,MAGNETIZED,HOMOGENEOUSSTAR No. 3,1969 6 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Equations(A3)are,ofcourse,notstrictlyvalidforneutron starswheregeneral-relativisticcorrec- For thepurposeofcalculatinggravitationalradiationfrom,orprecessionof,arotating M x3 3 ik =fBiBkPx(MSMii), jjr _Grp()p('i—x'i)(xx'k)tpXdx' w«- J/Ix-x'l k Pik -fpikPx. 2 Tik —21/*) 2 2 Pzz —2MzzMWzz0. P22 +/22Û-2M22MW22=0, (A8) P11 +/iiO-2^!MWn=0, 2-2M+WikPahikM, =Tik Dij =31ij—àij^Ikk.(A2) hj =J'pXiXjdPx,(Al) APPENDIX A PULSARS k (A7) (AS) (A6) (A4) (A3) 1411 19 6 9ApJ. . .157.13950 where Chandrasekhar andLebovitz(1962)haveshownthat\ In ordertoproceedfurther,wemustevaluatethe11’,;symbols.Forahomogeneousellipsoid we findthat,toordere, Now letting and which, withequation(All),gives We formallycompletethesolutionbygivingthree moments-of-inertia tensorintermsof€¿: The equatorialellipticity€requiredfordetermining the gravitationalradiation(cf.[5])is the ellipticitiese,-.Wefind Now wemaysubstituteequations(A13)and(A14)into(A9)(A10)tosolvefor 1412 J.P.OSTRIKERANDE.GUNNVol.157 How arewetoevaluatetheanisotropicpressureandmagnetic terms?Forthepurposeofillus- tration, considerFerraro's(1954)modelofahomogeneous magneticstarhavingasimplein- terior poloidalfieldwhichisfittedtoanexternaldipole acrossasurfacefreeofcurrents.Forthis c model onecanshowthat ' © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem ** =«2-«r[(P22Pu)2(3/22Mn)]. ■’(A18) 3 =2 —2 2 « IfF[-2(Mn-J/22)+(PuP22)/O]., í2 =¿[2(Jfii-JÍ22)+(Puh)-I/O],(Aló) =[4(Jin-M22)2(PuP22)I/O], Ai = ei W -=2(MnM)(PuP22),(A9) Wu -Wü=.(A10) n22 Wu -Wjj=tj) 21/ 0 («3+M)[(aiu)(aw)]‘ Wu -—-^GWaiAi(nosum), 23 ^6, =0,Ai(1—§€i) ” du la =i/(l—€¿)(hósum). (A17) ; : (Mn -Mn)poi= '(A19) 3 = —aaifhctz, w =— a 2 cm a (no sum), (All) (A12) (A15) (A13) (A14) 19 6 9ApJ. . .157.13950 2 13 2-9 2 15 An internaltoroidalmagneticfield,whichwouldnotbeobservableexceptforthedistortionit produces, might,onvariousgrounds,beexpectedtogreatlyexceedthepoloidalfield.Here where (B^)isthevolume-weightedaverageof(B+B).Theanisotropicpressureterm No. 3,1969 5 -4 15 Then, fromequation(A21)andthescalarvirialequations,onecanshowthat somewhat moredifficulttoestimate.Let out that,inamagnetizeddegenerateelectrongas,‘thepressuresparallelandperpendicularto We willconsideronepossiblecauseofpressureanisotropy.CanutoandChiu(1968)havepointed where RistheSchwarzschildradiusofstar.Thus,forthismechanism,pressureanisot- pressure (P)is where Bisthe“critical”field(£^4X10gauss),andxratiooflocalFermienergy the fieldaredifferent.In“low”limittheypredictthat,forelectrons, ropy (8)tobeUsedinequation(A21)is with B~Beyn^lO",and#3X10wefind{ô)tobeaboutthusnegligible. The equatorialeccentricitycannowbeevaluated;wehave to mc.Itiseasyshowthattheratioofaverageelectronpressure{P)total Although anisotropyintheelectronpressureisapparently negligible,thereexiststhedifficult yz ing thispossibility,wehave,consistentwithourprevious approximations, to evaluatepossibility(Ruderman1969),thattheneutron pressureitselfisanisotropic.Neglect- s c or en e for theCrabpulsar.Thusinteriorpoloidalfieldsof ~10 gaussorsomewhatlargertoroidal fields areneededtoproducetheeccentricityrequired for thegravitationalradiationadducedin § lib.Although3ordersof magnitudegreaterthanthesurfacefield,10gausswill, byderiva- field ofonly~10gausson themainsequence. tion, notsignificantlydistort thestar(e~3X10),andthisvaluecorresponds toaninterior e © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem = —1(^>+W-(B/)).(A26) (M22 ~Mn)tor=^(BJ),(A20) 2 Z62 ~(P) (Pe) ~f§^L(3P- W», (A27) € ~4X10~(3B—<£», fíp eP0 S(r) = P11-P22 =|<Ô)|TF|.(A22) (S)= (S)(P/(P).(A25) ele / PULSARS Pii(r) —P22O') / a\me\/«A \TlJ \mJ\n’ n SP«(r) (A21) (A23) (A24) 1413 19 6 9ApJ. . .157.13950 at constantrd.Herex//isthe fieldamplitudeand,ofcourse,isdeterminedatthe retarded time. we musthave fixed, losing kineticenergyonatimescalelongcomparedwiththeperiodofrotation,rateloss angular momentumis if Atisnottoolarge.Sincethe(¿»-dependenceforany wth-ordermultipoleisjust i.e., themomentsrotatewithangularfrequencyß. flux isthephasevelocity. following conditions. body necessarilycarriesangularmomentuminanamountgivenbyequation(Bl)underthe is justdE/dJforarigidconfiguration,which say oneofordern.Conditionbrequiresthatthefield beunchangedunderthetransformation tion directionisnormaltosurfacesofconstantphase,andtheratioenergyfluxmomentum ters whichdescribethefigureofbodybe(ai,.,a).Then,ifweconsidermassto safely beassumedtoproceedthroughasequenceofequilibriumconfigurations.Lettheparame- term ontherightremains.However, Each terminthesumvanishesforanequilibriumconfiguration,however,andsoonlyfirst this defines“slowly.” multipole momentofordernisgivenby and thedesiredresultfollowsimmediately. 1414 n 1 We wishtoshowfirstthat,foranystable,uniformlyrotating,equilibriumconfiguration © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The periodofrotationisnecessarilyaslongthedynamicaltimescale,soevolutioncan We nextwishtoshowthatmultipoleradiationemittedslowlybyanyuniformlyrotating Take apointinthefarwavezoneandconsiderwavesbelonging toonemultipolecomponent, a) Thefluxofradiationisproportionaltothesquarefieldamplitude\p,propaga- b) IfOa{)istherotationmatrixforaabout2-axisthroughanangle0, c) ß“dÜ/dt«ß; (B 6) ENERGY ANDANGULARMOMENTUMLOSSESOFAROTATINGSTAR D aß)=Oa'ßM)-...Oß(üt)Dö(Ö)f(ß),(B5) aiai K J. P.OSTRIKERANDE.GUNN , /->0+ßA/,tAt, (B7) \¡/ ~ (B9) t =Ae™*+Be-, (B8) APPENDIX B dJ dj).’ dE _dE\ a d¿ IdE = dt Ü' E =Jti,(B4) dJ + Vol. 157 (Bl) (B2) 19 6 9ApJ. . .157.13950 2-4 2 proof. rotational energyandangularmomentumslowly alwaysinthe“correct”ratio. moments areinstantaneouslyfixedinthebodyremains anequilibriumconfiguration,losing as desired.TheauthorsaregratefultoProfessorJ.A. Wheeler foradiscussionwhichledtothis Integrating 3IIandFoverasphere,weobtain radial fluxofangularmomentumparalleltoQis and, recallingthatFo(r,d)ocr~itiseasilyseenallothertermsareoforderr.Thusthe Thus thedominanttermis while theangular-momentumfluxisgivenby and themomentum-fluxtensoris Let Fbethemagnitudeofenergyfluxatr,0.Then and, normalizingtoorder1/r,wehave in generalhavetheform where thedependenceoíAontisweakinsenseofconditionc.Thepropagationvector where visthepropagationvelocity,andsofromequations(B9)(BIO)amplitudemust then proportionaltothegradientofphase; The dependenceonratanygiven0,0,however,isjust No. 3,1969PULSARS1415 0 7 n Putting thetworesultstogether,weseethatarotating equilibriumbodywhosemultipole It canbeshownthattheelectrical conductivityainsideneutronstarisvery roughly © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem =231 2*/*a(2tiv_ (CIO) rroé+ tded Pna* =P«(l+kAo), (C8) mx =mu.eixp(—t/’td)*(C5) 231 1/2 (T =0.6X10sec“(Ç2) F =P(//ro). (C9) ma3t0d 6 h & =4X10yr. 2 4aV TTC Vol. 157 (C4> (C3) 19 6 9ApJ. . .157.13950 -1 can Julian, W.H,andGoldreich,P1969,paperpresentedattheFourthTexasSymposiumonRelativistic period (dP/dt)o=Poroenablesustocalculatetheageofpulsar,maximumit initial field(B)isalwaysgreaterthan\ Hartle, J,andThorne,K.1968,Ap.153,807. The instantaneousvalueofthefield