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Mon.Not.R.Astron.Soc.000,1–10(2015) Printed2April2015 (MNLATEXstylefilev2.2) Rotational evolution of the Crab pulsar in the wind braking model 5 1 F. F. Kou and H. Tong 0 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China; [email protected] 2 r p A 2015.3v3 1 ] ABSTRACT E The pulsar wind model is updated by considering the effect of particle density and H pulsardeath.Itcandescribeboththeshorttermandlongtermrotationalevolutionof pulsarsconsistently.ItisappliedtomodeltherotationalevolutionoftheCrabpulsar. . h The pulsar is spun down by a combination of magnetic dipole radiation and particle p wind. The parameters of the Crab pulsar, including magnetic field, inclination angle, - o andparticledensityarecalculated.Theprimaryparticledensityinaccelerationregion r is about 103 times the Goldreich-Julian charge density. The lower braking index best tweenglitchesisduetoalargeroutflowingparticledensity.Thismaybeglitchinduced a magnetosphericactivities innormalpulsars.Evolutionof brakingindex andthe Crab [ pulsar in P −P˙ diagram are calculated. The Crab pulsar will evolve from magnetic 2 dipole radiation dominated case towards particle wind dominated case. Considering v theeffectofpulsar“death”,theCrabpulsar(andothernormalpulsars)willnotevolve 4 to the cluster of magnetars but downwards to the death valley. Different acceleration 3 modelsarealsoconsidered.Applicationsto othersourcesarealsodiscussed,including 5 pulsars with braking index measured, and the magnetar population. 1 0 Key words: pulsars:general–pulsars:individal(PSRB0531+21)–stars:magnetar . – stars: neutron – wind 1 0 5 1 v: 1 INTRODUCTION ing index. The braking index and second braking index are i definedrespectively (Livingstone et al. 2005): X The Crab pulsar (PSR B0531+21) is a young radio pulsar with a spin frequency ν =30.2Hz and frequency derivative νν¨ r n= , (2) a ν˙ = −3.86×10−10Hz/s (Lyne et al. 1993). Its character- ν˙2 istic magnetic field is about 7.6×1012G at the magnetic ν2 .ν.. poles1. The Crab pulsar has been monitored continuously m= ν˙3 , (3) for decades years by various telescopes since it was discov- ... where ν is the pulsar spin frequency, ν˙, ν¨ and ν are the ered in 1968 (Lyne et al. 1993; Lyne et al. 2015; Wang et first, second and third frequency derivative, respectively. al. 2012). Its braking index is n = 2.51±0.01 (Lyne et al. The spinning down of pulsars is usually assumed to be 1993). Different braking index values 2.45, 2.57 and 2.3 are brakedbythemagneticdipoleradiation(Shapiro&Teukol- also reported between glitches (Wang et al. 2012; Lyne et sky1983).Inthemagneticdipoleradiationmodel,aneutron al. 2015). Observational details are given in Table 1. star rotates uniformly in vacuo at a frequency ν and pos- Long-termobservationshavefoundthatalmostallpul- sesses a magnetic moment µ. The corresponding slowdown sars (except accreting X-ray pulsars in binary systems) are rate is: spinningdown.Thespin-downbehaviorcanbedescribedby a power law (Lyneet al. 1993): ν˙ =−8π2µ2ν3sin2α, (4) 3Ic3 ν˙ =−kνn, (1) whereµ=1/2BR3 isthemagneticdipolemoment(Bisthe here, k is usually taken as a constant and n is the brak- polar magnetic field and R is the radius of neutron star), I = 1045g·cm2 is the moment of inertia, c is the speed of light,andαistheanglebetweentherotationalaxisandthe magnetic axis (i.e., the inclination angle). The spin down 1 Assuming all the rotational energy loss is due to magnetic behavior of pulsar in this model can be described as ν˙ ∝ dipoleradiationinvaccum,Bc=6.4×1019pPP˙G ν3. The braking index is exactly three if µ, I, and α are 2 Kou & Tong constant. To date, only eight pulsars, including the Crab Table 1.ObservationsoftheCrabpulsar. pulsar, have measured the meaningful braking indices for they are young and own relatively larger ν˙ (Espinoza et al. Pulsarparameter Values 2011;Lyneetal.2015).Theirbrakingindicesareallsmaller thanthree.Itmeansthatthereareotherphysicalprocesses Epoch(MJD) 40000.0 needed to slow down the pulsar. ν(Hz) 30.225437a Many mechanisms have been proposed in order to ex- ν˙(10−10Hz/s) −3.86228a ν¨(10−20Hz/s2) 1.2426a plain the braking index observations, e.g., the pulsar wind .ν..(10−30Hz/s3) −0.64a model (Xu & Qiao 2001; Wu et al. 2003; Contopoulos & Brakingindexn 2.51±0.01a Spitkovsky2006;Yueetal.2007),achangingmagneticfield Secondbrakingindexm 10.15a strength (Lin et al. 2004; Chen & Li 2006; Espinoza et al. Inclinationangleα(◦) (45∼70)b 2011), a changing inclination angle (Lyne et al. 2013), and Age 915yr(from1054to1969) additional torques due to accretion (Liu et al. 2014). How- ever,thesemodelsaremoreorlessnotconsistentwithobser- Notes:(a):Observedspinfrequencyanditsderivativesare vations or can not simulate the long-term evolution of pul- parametersintheTaylorexpansion sars.Andthesemodelscannotexplainthedifferentbraking ν=ν0+ν˙0(t−t0)+ 12ν¨0(t−t0)2+ 16 ν..0. (t−t0)3 for indices detected between glitches (Wang et al. 2012; Lyne 1969∼1975(Lyneetal.1993). Fortherearerareglitches et al. 2015). Furthermore, the effect of pulsar death (death occurredinthisinterval.Later observations(Lyneetal.2015) line, Ruderman & Sutherland 1975; or death valley, Chen confirmpreviousresults. &Ruderman1993; Zhanget al. 2000) shouldbeconsidered (b):Therangeofinclinationangleisgivenbymodelingthepulse profileoftheCrabpulsar(Dyksetal.2003; Hardingetal.2008; inmodelingthelongtermrotationalevolutionofthepulsar Watter etal.2009;Duetal.2012; Lyneetal.2013). (Contopoulos & Spitkovsky2006). The pulsar wind model considers both the pulsar spin- down and the particle acceleration in the magnetosphere ontheacceleration potentialdrop.Thecorresponding rota- (Xu&Qiao2001;Wuetal.2003).Inthispaper,anupdated tional energy loss rate is (Xu & Qiao 2001): pulsarwindmodelisbuiltbasedonpreviousresearches(Xu & Qiao 2001; Yue et al. 2007; Contopoulos & Spitkovsky E˙p=2πrp2cρe∆φ, (6) 2006; Li et al. 2012). It includes: (1) Both magnetic dipole wherer =R(RΩ/c)1/2ispolargapradius,ρ =κρ isthe radiation andparticleoutflow,andtheirdependenceonthe p e GJ primaryparticledensity(ρ =ΩB/(2πc)istheGoldreich- inclination angle; (2) The particle outflow depends on the GJ Julian charge density,Goldreich & Julian 1969 ), ∆φ is the specificaccelerationmodel;(3)Theprimaryparticledensity corresponding acceleration potential of the acceleration re- may be much larger than the Goldreich-Julian charge den- gion,andκisacoefficientrelatedtoprimaryparticledensity sity;(4)Theeffectofpulsardeathisconsideredinmodeling which can be constrained by observations2. The maximum therotationalevolutionofthepulsar.Thismodelcancalcu- acceleration potential for a rotating dipole is ∆Φ=µΩ2/c2 lateboththeshorttermandlongtermevolutionofpulsars. (Ruderman & Sutherland 1975). Then equation (6) can be ItisappliedtotheCrabpulsarwhichhasthemostdetailed rewritten as (considering that the particle acceleration is timing observations. Possible applications to other sources mainly related to the parallel component of magnetic mo- are also illustrated. ment): The pulsar wind model and model calculations of the Crabpulsararepresentedinsection2.Discussionsandcon- 2µ2Ω4 ∆φ E˙ = 3κ cos2α. (7) clusions are given in section 3 and section 4, respectively. p 3c3 ∆Φ The magnetic dipole radiation and the outflow of particle wind may contribute independently. Then the total rota- 2 SPIN-DOWN OF THE CRAB PULSAR IN tional energy loss rate is: THE PULSAR WIND MODEL 2µ2Ω4 ∆φ 2.1 Description of the pulsar wind model E˙ = E˙d+E˙p = 3c3 (sin2α+3κ∆Φcos2α) Inthepulsarwindmodel,therotationalenergyisconsumed 2µ2Ω4 = η, (8) bymagneticdipoleradiationandparticleacceleration(Xu& 3c3 Qiao 2001). The magnetic dipole radiation is related to the where η = sin2α+3κ∆φ/∆Φcos2α. The first and second perpendicular component of the magnetic moment (µ⊥ = items of expression η are respectively for magnetic dipole µsinα), the power of magnetic dipole radiation is (Shapiro andparticlewind.Iftheaccelerationpotential∆φ=0,there & Teukolsky 1983): are no particles accelerated in the gap, the pulsar is just 2µ2Ω4 braked down by the magnetic dipole radiation. The accel- E˙d = 3c3 sin2α, (5) eration potentialis modeldependent(Xu&Qiao 2001; Wu et al. 2003). A particle density of κ times Goldreich-Julian whereΩ=2πν istheangularspeedofthepulsar.Theeffect ofparallelcomponentofmagneticmoment(µ =µcosα)is k responsible for particles acceleration (Ruderman & Suther- 2 Primary particle density in the acceleration gap is: ρe = land 1975). Acceleration gaps are formed above the polar |e|[n++n−], but Goldreich-Julian “charge” density is: ρGJ = gap,primaryparticlesaregeneratedandacceleratedinthese |e|[n+ −n−] (Yue et al. 2007). It is reasonable to infer that: gaps. The energy taken away by these particles dependents κ≥1. Rotational evolution of the Crab pulsar in the wind braking model 3 charge density is considered. Expressions of η are given in Table2.TheExpressionsofηfornineparticleaccelerationmod- Table2forvariousaccelerationmodels3.Therotationalevo- els. lution of the pulsar can be written as: −IΩΩ˙ = 2µ2Ω4η, (9) No. Accelerationmodel η 3c3 1 VG(CR) sin2α+4.96×102κB1−28/7Ω−15/7cos2α According equation (2) and (3), thebraking index and sec- 2 VG(ICS) sin2α+1.02×105κB1−222/7Ω−13/7cos2α ond braking index in the wind braking model are: 3 SCLF(II,CR) sin2α+38κB1−21Ω−7/4cos2α 4 SCLF(II,ICS) sin2α+2.3κB1−222/13Ω−8/13cos2α Ω dη 5 SCLF(I) sin2α+9.8×102κB1−28/7Ω−15/7cos2α n=3+ η dΩ, (10) 6 OG sin2α+2.25×105κB1−212/7Ω−26/7cos2α 7 CAP sin2α+54κB1−21Ω−2cos2α m=15+12Ω dη +(Ω dη)2+ Ω2 d2η. (11) 89 NNTTVVGG((CICRS)) ssiinn22αα++1639..76βγ−0.114κκBB1−21−121ΩΩ−−1.18.876cocso2s2αα η dΩ η dΩ η dΩ2 Notes:Particledensityρe=κρGJ istakeninallthesemodels.B12 isthe Equation (9), (10) and (11) are all functions of magnetic magneticfieldstrengthinunitsof1012G. fieldB,inclinationangleαandparticledensityκ(Ωandits (a):Themodels1∼5arebasedonthepulsarwindmodelofXu&Qiao (2001).TheVGandSCLFarerespectivelytheaccelerationmodels: derivativesare observational inputs, see Table 1). vacuumgapandspacechargelimitflow.Inthevacuumgap(Ruderman& Taking these expressions of η (Table 2) back to equa- Sutherland1975),curvatureradiation(CR)andinverseCompton tion(9),itcanbeseenthat:asthepulsarspinningdown(Ω scattering(ICS)areconsidered(Zhangetal.2000).IntheSCLFcase, decreases), the effect of particle wind becomes stronger. If regimesIIandIaredefinedbyfieldsaturatedornot(Arons& Scharlemann1979;Harding&Muslimov1998).Threemodelsare there are particles flowing out, it means that there are still introduced:CRforSCLFmodelinregimeII,ICSforSCLFmodelin particleaccelerationinmagnetosphere(thepulsarisstillac- regimeII,andtheSCLFmodelinregimeI. tive).However,aspulsar spinningdown, thepotential drop (b):TheOGmeanstheoutergap,isself-sustainingandlimitedbythe maynotsustaintheneedtoaccelerateparticles(Ruderman electron-positronpairproducedbycollisionsbetweenhigh-energyphotons (Zhang&Cheng1997).ThemodificationbyWuetal.(2003)isadopted. & Sutherland 1975). Besides, observations do not support (c):TheCAPmodelisaphenomenologicalmodelofconstantpotential pulsars to evolve unceasingly in these pulsar wind models ∆φ=3×1012V(Yueetal.2007). (Youngetal.1999;Contopoulos&Spitkovsky2006).There- (d):TheNTVGshortsfornearthresholdvacuumgap.Inthismodel,the fore,“pulsardeath”mustbeconsideredinmodelingthelong electron-positronpairproducedatornearthekinematicthreshold term rotational evolution of the pulsar. As the spin angu- h¯ω=2mc2/sinθbecauseofsuperstrongsurfacemagneticfield(Gil& lar speed decreasing to the death value, the radio emission Melikidze2002).β=52istakenintheCRcase,andγ=14istakenin tendstostop.Andwhenpulsarisdeath,thepulsarisbraked theICScase(Abrahams&Shapiro1991;Wuetal.2003). onlybymagneticdipoleradiation.Here,theVG(CR)model is employed to show the effect of pulsar death by introduc- ingapiecewisefunctiondeduction(detailsareshowninthe 2.2 Understanding the braking index of the Crab appendix A). pulsar  sin2α+4.96×102κ(1−ΩdeΩath)B1−28/7Ω−15/7cos2α, PTahreaVmGet(eCrsRo)fmthoedeClriasbtapkuelnsaarsaarenceaxlacumlaptleedtoinshthowisstehceticoanl-. ηVCGR= sini2fαΩ,>Ωdeath (12) culation process. For pulsars with the observed ν, ν˙, ν¨ and ifΩ<Ωdeath. .ν.., their observational braking index n and second braking  index m can be get byequation (2) and (3).Pulsar param- The death period is defined as Ω = 2π/P (Con- death death eters such as magnetic field, inclination angle, and particle topoulos & Spitkovsky2006; Tong & Xu2012): densitycan becalculated bythesethreeequations(9),(10) B V and (11). The primary particle density may be 103 ∼ 104 P =2.8( )1/2( gap )−1/2s. (13) death 1012G 1012V times of Goldreich-Julian charge density (Yue et al 2007). According the observational range of inclination angle and V can be viewed as the maximum acceleration poten- gap the characteristic magnetic field (which can be viewed as tial drop in the open field line regions and V = 1013V gap order of magnitude estimation of the true magnetic field), is chosen in the following calculations (Contopoulos & the range of particle density κ can be limited by equation Spitkovsky2006).ItisintroducedaccordingtoContopoulos (9) and (10). In our calculation, κ = 103 is adopted4, the &Spitkovsky(2006)(seealsolaterworksbyLietal.2012). inclination angle α andmagnetic field B areabout 55◦ and In this way, the effect of pulsar death can be incorporated 8.1×1012G which are consistent with observational con- intherotationalenergylossrate.Theeffectofpulsardeath straints. The second braking index calculated by equation is discussed in more detail in section 2.5. Foryoungpulsars (11) is 10.96, which is roughly consistent with theobserved (Ω≫Ω ),theeffectofpulsardeathisnegligible.Butfor death value10.15(Lyneetal.1993),consideringtheobservational the long-term evolution of pulsars, especially when the pe- uncertaintities (Lyneet al. 1993, 2015). riod approachesthedeathperiod, theeffect ofpulsardeath The magnetic field and inclination angle are assumed is significant. to be constant in the pulsar wind model. The particle densitydecreases inlong-term evolutionof pulsarsbutremains unchanged in short duration at early time. Figure 1 shows the braking index of the Crab pulsar as a function of spin 3 AsshowninTable2,themodels ofVG(CR)andSCLF(I) are verysimilarwitheachother.Crabparameterscalculatedinthese 4 We should note that the κ is related to the primary particles twomodelsarealsosimilar(shownlater). intheaccelerationgapbutnotthetotal out-flow particles. 4 Kou & Tong 3.0 2.3 An increasing particle density results in a lower braking index during glitches 2.5 Abrakingindex2.3hasbeenmonitoredaftertheremovalof theeffectsofglitchesduringan epochwhentherearemany x de glitches occured (Lyne et al. 2015). It is different from the n gi 2.0 longtermunderlyingbrakingindex2.51oftheCrabpulsar. n aki Previously, Wang et al. (2012) have measured two different br values of braking index n = 2.45 and n = 2.57 between 1.5 glitches. They attribute it to the effect of varying particle wind strength (Wang et al. 2012). It is generally accepted that glitch is caused by the inner effect but may lead to 1.0 some effect in the outer magnetosphere5, for example: the 0 20 40 60 80 100 changing of primary particle density during this epoch. In PHmsL the pulsar wind model, it can be denoted by the changing Figure 1. The braking index of the Crab pulsar as function of coefficient of particle density κ = κ(t). Then equation (10) spin period in the VG(CR) model. The dashed line is observa- should be modified: tional brakingindex of 2.51. The dotted lineisbraking index in Ω dη κdηΩκ˙ thetransitionpoint(n=2). n=3+ + . (14) η dΩ ηdκΩ˙ κ Equation (14) can be rewritten as: Ω dη κdη τ n=3+ − c, (15) Table 3. Parameters of the Crab pulsar calculated in various η dΩ ηdκτ κ accelerationmodels. where τ = − Ω is the characteristic age, τ = κ can be c 2Ω˙ κ 2κ˙ viewedasthetypicaltimescaleofthechangingparticleden- Models κ α B12 P0 Pt Pd Tt Td sity. When the magnetosphere is in equilibrium (i.e., when 103 ◦ ms ms s 103yr 105yr the glitch activities of the pulsar is not very active), the VG(CR) 1 55 8.1 18.8 55 2.52 2.77 0.78 time scale of particle density variation (τκ) may be very VG(ICS) 0.1 60 7.5 18.9 63 2.58 3.50 1.67 large (i.e., larger than the characteristic age τ ). The third SCLF(IICR) 2 58 7.5 18.9 68 2.59 3.93 2.24 c SCLF(IIICS) 0.6 44 5.0 19.2 − 3.16 − 30.1 term in equation (15) does not contribute. But, if they are SCLF(I) 0.6 57 7.8 18.8 55 2.45 2.77 0.78 comparablewitheachother,thebrakingindexchangessub- OG 20 59 8.2 18.5 42 2.47 1.63 0.07 stantially.Inotherwords,whentheoutflowparticledensity CAP 3 52 8.3 18.9 58 2.46 3.08 1.14 NTVG(CR) 3 57 7.7 19.5 67 2.56 4.10 2.52 increases and the effect of particle wind becomes stronger, NTVG(ICS) 30 54 7.5 19.4 62 2.58 3.58 1.81 means τ > 0 or κ˙ > 0, the braking index will be smaller κ than 2.51. But when the outflow particle density decreases Notes: (τ < 0 or κ˙ < 0), the braking index will be larger than (a):κisthecoefficientofprimaryparticledensityρe=κρGJ. κ (b):αandB12 arerespectivelytheinclinationangleandthemagnetic 2.51. As the out-flow particle density tending to a certain field. value (larger or smaller than previous value), the braking (c):P0istheinitialperiod. index tends to its underlying value (2.51). Generally, the (d):PtandTt arethetransitionperiodandcorrespondingage.Atthis transitionpointthebrakingindexis2.Becausetheminimumbraking increased (or decreased) component of the outflow particle indexofmodelSCLF(ICS)is2.4,thereisnosuchatransitionpoint, density (may be 0.1%) is so much small that it can be ig- whichcanbeseenintheevolutioninP−P˙ diagram(seeFigure7). (e):PdandTdaretheperiodandagewhenpulsarstopsradioemission. nored. Thedeathperiodiscalculatedbyequation(13). Figure 2 shows the braking index as function of τ for κ the Crab pulsar in the VG(CR) model. In order to bet- ter understand the observations of n < 2.51 (Lyne et al. 2015), we consider the τ > 0 only. As we can see in this κ figure,thebrakingindexisinsensitivetoτ whenitislarger κ period. As the pulsar period increases, the braking index than104yr.But,when it iscomparable with thecharacter- decreases from 3 to 1. It illustrates that the Crab pulsar is isticage(about103yr),thebrakingindexdecreasessharply. initially braked by magnetic dipole (n → 3) and then by When braking index is 2.3, the changing rate of the parti- the particle wind (n→1). We take n=2 as the transition cledensityisκ˙ ≈1.2×10−8s−1.Inthisinterval(fromMJD point(seeFigure5,atn=2,theslop ofthecurveis0)and 51000toMJD53000,about5years),theparticledensityhas the corresponding spin period is about 55ms. Since Table changed by2ρGJ.Thechangeof particle densitywill result 1 has shown the timing observations of the Crab pulsar in in changes of radiation energy loss rate and period deriva- 1969 (its age is 915 years old by then), its initial period is tive.Theirchangesrespectivelyare:δE˙ ≈2×1035erg/sand P ≈ 18.8ms in AD 1054 by integrating equation (9). The δP˙ ≈1.88×10−16s/s. 0 calculationsofallacceleration modelsareshowninTable3. As shown in this table, the particle densities are different in different models, butare all more than 100 times of ρ . GJ Theinclinationanglesiswithintheobservationalrange.The 5 Glitch induced magnetospheric activities are very common in initial periods are all about 19ms. thecaseofmagnetars (Dib&Kaspi2014). Rotational evolution of the Crab pulsar in the wind braking model 5 3.0 • 4.225´10-13 • brakingindex 122...505 22.5.31 H(cid:144)LPeriodderivativess 444...444122...4901122.5559122´´´´´´´11111110000000-------11111113333333 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 0.0333 0.0334 0.0335 0.0336 0.0337 1.0 PeriodHsL 100 200 500 1000 2000 5000 1´104 ΤkHyrL Figure 3. Evolution of the Crab pulsar in P −P˙ diagram in VG(CR) model compared with observations. The black points Figure2.ThebrakingindexoftheCrabpulsarasfunctionofτκ are timing results from Jodrell Bank monthly ephemeris. The intheVG(CR)model.Thedashedlineistheunderlyingbraking blue points are the rotational evolution of Crab pulsar in the index2.51.Thedottedlineisthesmallerbrakingindex2.3mea- VG(CR)model.Theredonesareobtainedbyaddingtheeffectof suredduringglitchactivities(Lyneetal.2015,Figure7there). glitches(Lyneetal.2015,Table3there).Andthegreenonesare thesummationof redpoints andafactor δP˙ ≈1.88×10−16s/s (from MJD 51804.75 afterwards), which may be caused by the Thepulsarwindmodelcanbetestedwithobservations change of out-flow particledensity. Atfirst,the black, blue, red, (JodrellBankmonthlyephemerisoftheCrab pulsar6).The andgreenpointsarecoincidewitheachother.(Notes:Thefitting effect of glitches is not taken into consideration in the pul- doesnotmodelthetransientprocesscausedbyglitches.) sar wind model. The changes of period and period derivative caused by glitches should be added (Lyne et al. 2015). evolves with period by a function of P˙ ∝ P2−n (Espinoza Figure 3 is the model calculation compared with observa- et al. 2011). In the log-log plot, the evolution curveevolves tions (without modelling the transient variation caused by withaslop(2−n).Asthebrakingindexfallingfrom3to1, glitches).Whentheamplitudeofglitchesisrelativelysmall, thepulsarevolvefrommagneticdipoleradiationdominated thechangeofparticledensityisnotobvious,thenthemodel case (the left points) to the particle wind dominated case calculations(thebluepoints)fitwellwithobservations(the (the right points) and the bottom is the transition point black points). However, when the amplitude of glitches is (n = 2). Clearly, the curve evolves more sharply after the large, the effect of glitches must be added (the red points). transition point.The asymptoticbehaviors are discussed in Moreover, the effect of particle density change should be taken into consideration. A factor δP˙ = 1.88 ×10−16s/s theappendix B. (whichiscalculated above)isaddedtotheredpoints(from MJD 51804.75 afterwards), shown as the green points. The 2.5 The effect of pulsar death greenpointscanmatchthegeneraltrendoftheobservations. As shown is the Figure 5, the line evolves up right with a slopeabout1afterthetransitionpointjustlikePSRJ1734- 2.4 Long term evolution of the Crab pulsar 3333whosebrakingindexis0.9±0.2(Espinozaetal.2011). If the pulsar continually evolve in this trend, it would not Thepulsarperiodanditsfistderivativeevolutionwithtime dead and might end itself as a magnetar. It will be hard in the VG(CR) model are shown in the first and second to explain the observations for: (1) The number of magne- panel of Figure 4. The curves evolve slowly in its early age tars are smaller than radio pulsars and most of the pulsars and then rise sharply, indicating that the Crab pulsar is occupy the central region in the P-P˙ diagram (see Figure mainlybrakedbymagneticdipoleradiationfirstlyandthen 6);(2)Onlyasmallportionofmagnetarsareradioemitters mainlybyparticlewind.Thetransition period is55msand (Olausen & Kaspi 2014); (3) The properties of the Crab ageoftheCrabpulsarwillbe2771calculatedintheVG(CR) pulsar, PSR J1734-3333 and other high magnetic field pul- model(seeTable3).Thebrakingindexevolutionwithtime sars are significantly different from magnetars (Ng& Kaspi in theVG(CR) model is shown in thethird panel of Figure 2011).Aspulsarsslow-down,thedensityofoutflowparticles 4. Braking index decreases from 3 to 6/7 –different mini- maydecreaseandtheeffectofparticlewindwillrecede(for mum braking indices for different acceleration models. The the SCLF cases, the number of particles (or κ) may close bottom panel of Figure 4 shows the second braking index to constant, but the particle density still decreases because evolution with time. Thecurvegradually decreases from 15 a reducing Goldreich-Julian charge density). Therefore, the to 30/49. Figure 5 shows the evolution of the Crab pulsar in P −P˙ diagram from birth to its 30000 years old. effectofpulsardeathmustbeincludedinmodelingthelong- term rotational evolution of pulsars. These points are respectively taken when the Crab pulsar Theessentialconditionofradioemission forapulsaris is:1,915, 2771, 9000and30000yearsold.Period derivative itspairproduction (Sturock1971; Ruderman&Sutherland 1975). The so called pulsar “death” means the stopping of 6 http://www.jb.man.ac.uk/pulsar/crab.html, data from 1982 pulsaremission(Ruderman&Sutherland1975;Chen&Ru- February 15 (MJD 45015) to 2014 October 15 (MJD 56945) are derman1993; Zhangetal. 2000; Contopoulos &Spitkovsky usedhere. 2006).ThedeathdefinedbyRuderman&Sutherland(1975) 6 Kou & Tong 1.5´10-11 2.0 1.0´10-11 7.0´10-12 HLPeriods 11..05 Periodobservation H(cid:144)Lerivativess 523...000´´´111000---111222 dd 1.5´10-12 o 0.5 eri 1.0´10-12 P 0.0 0 5000 10000 15000 20000 25000 30000 0.02 0.05 0.10 0.20 0.50 1.00 2.00 ageHyrL PeriodHsL 1.4´10-11 1.2´10-11 Figure5.EvolutionoftheCrabpulsarintheP-P˙ diagram.Five H(cid:144)Lvess 1.´10-11 ypeoainrstsoaldre.chosenwhenthepulsaris:1,915,2771,9000and30000 derivati 68..´´1100--1122 Periodderivativeobservation d o Peri 4.´10-12 isthat:apulsardeadwhenthemaximumpotentialdrop∆Φ 2.´10-12 available from the pulsar is smaller than the acceleration potential needed to accelerate particles. Because this death 0 0 5000 10000 15000 20000 25000 30000 lineismodeldependent,Chen&Ruderman(1993)proposed ageHyrL death “valley” by modifying the boundary conditions. And this death “valley” was updated by considering different 3.0 particleacceleration andphotonradiationprocesses(Zhang et al. 2000). However these works are just to separate pul- 2.51 2.5 sarswithradioemissionfromthosewithout.Contopoulos& Spitkovsky (2006) included the effect of pulsar death when dex 2.0 modeling the rotational evolution of pulsars. However, the n gi braking indices there are always larger than three. Consid- n braki 1.5 ering particle acceleration and pulsar death simultaneously maygiveacomprehensiveinterpretationforbothshortand 1.0 long-term rotational evolution of pulsars. 6(cid:144)7 The visual explanation for pulsar death in the pulsar wind model is that the particles in the magnetosphere are 0.5 10 100 1000 104 exhausted and there are no primary particles generated. tHyrL For the physical understanding, we can refer to the equation (8), expressions ∆Φ ∝ Ω2 and ∆φ (i.e., ∆φ = VGCR 14 2.8×1013R2/7B−1/7Ω−1/7V, Xu & Qiao 2001), with the 6 12 pulsarspinningdown,themaximumpotentialofthepulsar 12 (∆Φ)drops,whenthemaximumpotentialcannotmeetthe dex 10 10.15 need to accelerate particles (∆φ > ∆Φ), the pulsar death. n i g Besides, the acceleration potential is weakly dependent on brakin 8 theΩ,that iswhywecan makecalculations underthecon- nd 6 stant acceleration potential (the CAP model) (Yue et al. o sec 4 2007). Figure 6 shows thelong-term evolutions of the Crab pulsar in the VG(CR) model. The dashed curve is the line 2 30(cid:144)49 showninFigure5andthesolidistheonewithpulsardeath (according to equation (12)). These two lines evolve simi- 0 10 100 1000 104 lar at the early age, and divide obviously after the transi- ageHyrL tionpoint.Thesecondlinefallsdownafterwardsandmoves Figure 4.Rotational evolutionoftheCrabpulsarfrombirthto down-right with a slop about −1 finally. The initial period 30 thousands years in the VG(CR) model. The first and second andtransitionperiodcanbecalculatedaccordingthesecond panelsarerespectivelytheperiodandperiodderivativeevolution line, which are same with the values given by the first line withtime.Thepointsareobservationsofperiodanditsderivative within precision. Indicating that the effect of pulsar death (Lyne et al. 1993). The third and fourth panels are respectively can be ignored in the early time. As mentioned before, the thebrakingindexandsecondbrakingindexevolutionwithtime. particle densitydecreases as pulsarspin-down. Thegap be- The dashed lines are observed values and the solid lines are re- tween these two lines is caused by the reducing of particle spectivelytheminimumbrakingindexandsecondbrakingindex density. When the spin period of Crab approaches to the intheVG(CR)model. death period 2.52s, the radio emission tends to stop and the slop of the second line is very large. Crab evolution in Rotational evolution of the Crab pulsar in the wind braking model 7 10-8 10-8 à(cid:144)ámagnetar øRRAT à OG NTVGICS à VGCR SCLFCR 10-10 ìæXCDCIONS ò inèteRrmRPittent pulsar áà àáàààààà 10-10 VCGAICPS NSCTLVFGICCRS áà àáàààààà H(cid:144)LPeriodderivativess 1111100000-----2111108642 Τæcææ11=æææ00æææææ169ææææææææ0æææææææyyææ3æææææææææææærrææææææææææææææææææyæææææææææææææærææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøæææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææòææææææææøææææææææææòæææææææææææææææææææææææææææææææææææææææææøæææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææædææææææææææææææøææææææææææææææææææææææææææææææææææææææòææææææææææææææææææææææeææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøæòæææææææøææææææææææaæææææææææææææææææææææææææææææææææææææææææææææææææææææætææææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææhææææææææææææææææææææææææææææææææææææææææææææææøæææææææææææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææælææææææææææææææææææææææææææææææiææææææææææææææææææææææææææææææææææææææææææææææææææænææææææææææææææææææææææææææææææææææææææææøææææeææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææàæææææææøææææææææøæææææææææææææææææææææææææææææææææææææìææææææææææææáæææøææææøøææøæææææææææææææáøæøàøææøæææìàææøàæBìææààæàæìcæì1àæìà=0æ110102G1140GG H(cid:144)LPeriodderivativess 1111100000-----2111108642 Τæcææ11=æææ00æææææ169ææææææææ0æææææææyyææ3æææææææææææærrææææææææææææææææææyæææææææææææææærææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøæææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææòææææææææøææææææææææòæææææææææææææææææææææææææææææææææææææææææøæææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææædææææææææææææææøææææææææææææææææææææææææææææææææææææææòææææææææææææææææææææææeææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææøæòæææææææøææææææææææaææææææææææææææææææææææææææææææææææææææææææææææææææææææætæææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææhææææææææææææææææææææææææææææææææææææææææææææææøæææææææææææææææææææææææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææælææææææææææææææææææææææææææææææiææææææææææææææææææææææææææææææææææææææææææææææææææænææææææææææææææææææææææææææææææææææææææææøææææeææææææææææøææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææàæææææææøææææææææøæææææææææææææææææææææææææææææææææææææìææææææææææææáæææøææææøøææøæææææææææææææáøæøàøææøæææìàææøàæBìææààæàæìcæì1àæìà=0æ110102G1140GG 10-22 10-22 10-3 10-2 10-1 100 101 10-3 10-2 10-1 100 101 PeriodHsL PeriodHsL Figure6.RotationalevolutionoftheCrabpulsarintheVG(CR) Figure 7. Rotational evolution of the Crab pulsar in all accel- model.Thedashedcurveistheevolutionwithoutconsideringpul- erationmodels.BecausetheVG(CR)modelandSCLF(I)model sardeath.Thesolidlineistheoneconsideringpulsardeath.The are very similar, their lines are coincident. Only the line in the P −P˙ diagram has listed all known radio pulsars, magnetars, VG(CR)modelisplotted. SeealsoFigure6forexplanations. andmillisecondpulsars(updatedfromFigure1inTong&Wang 2014).Thedot-dashedlinerepresentsthefiducialdeathline(Ru- derman & Sutherland 1975). Pulsars with meaningful braking wind model, the pulsar is braked down by magnetic dipole indices (Espinoza et al. 2011; Lyne et al. 2015) are defined by radiation and particle wind. But in the electric current loss largesquare.Thearrowsrepresenttheirevolutiondirectionsand model,theelectriccurrentflowgeneratesaspin-downtorque eacharrowindicateitsmotioninthenext10000 yr(FortheVela (Hardingetal.1999;Beskinetal.2007).Basedonthesame pulsar, 20000 yr is used; 5000 yr for PSR J1846-0258 and PSR magnetosphericphysics,thepulsarwindmodelandthecur- J1119-6127). rent loss case are just different point of views, will produce the same results (e.g. Section 3.2 in Tong et al. 2013). And various acceleration models are given in Figure 7. The bot- theycanbeusedtocheckeachother.Itismeaningfultotake tom line is theevolution curvein SCLF(ICS) model. There advantageof theexisting information and explain more ob- is no transition point, indicating that the particle wind ef- servations. The particle wind worked on pulsar spin-down fectofthismodelisveryweak.Thetoplineistheevolution wasfirstprovenbyobservationsofintermittentpulsarPSR oftheCrabpulsarintheOGmodel.Thismodelisforhigh- B1931+24 whoseradioemission switchesbetween“on”and energy radiation, the death of this model may differ from “of”state(Krameretal.2006). Correspondingcalculations radio emission models. The results here for the OG model forintermittentpulsarsinpulsarwindmodelweremadeby are just crude approximations to show the effect of pulsar Li et al. (2014). death.EvolutionoftheCrabpulsarinothermodelsaresim- As one of the best observed pulsar, observations of the ilar with each other. Crabpulsararerelativelycomprehensive(seeTable1&timing results from Jodrell Bank website). It will be the best target to test radiation models. The pulsar parameters can becalculatedbyequations(9),(10)and(11)whenmodelde- 3 DISCUSSIONS pendentacceleration potential (or η) is given (see Table 2). Inthemagneticdipoleradiationmodel,apulsarisassumed TheGoldreich-Julianchargedensityistakenastheprimary as a magnetic dipole rotating in vacuum. It is just a low- particledensityinthesemodelsofXu&Qiao(2001)andWu order approximation of the true magnetosphere, and there etal(2003).However,Yueetal.(2007)constrainedthepri- mustbeparticlesaccelerationandgenerationprocesssothat maryparticledensitybyobservationsofseveralpulsarsand the radio emission can be received. The actual magneto- predicted that it should be 103 ∼ 104 times of Goldreich- sphere surrounding the pulsar must be filled with plasma Julian charge density.In our calculations, the primary par- andcorotatewithpulsarbecauseofmagneticfreeze.Accord- ticledensityisatleast100timesofGoldreich-Julian charge ingthespecial relativity,thecorotation magnetosphere can density.IntheVG(CR)model,aparticledensity103|ρ |is GJ not infinitely extend but only exist within the light cylin- chosen. When applying the pulsar wind model to intermit- der. The magnetosphere is divided into two parts by the tent pulsars, a Goldreich-Julian charge density is assumed light cylinder: the closed magnetic field lines and the open (Lietal.2014).Asthepulsarspin-down,itsout-flowparticle magnetic field lines (including polar gap and outer gap). In densitydecreases. IntheP−P˙ diagram, theseintermittent the open field line region, particles are accelerated, gener- pulsars are on the right of those young pulsars (see Figure ate radiation and flow out (thus taken away some of the 6). This may explain the different particle density in the rotational energyof thecentralneutronstar). Inthepulsar Crab pulsar and intermittent pulsars. 8 Kou & Tong Other studies also found the need for a high particle braking index of PSR J1846−0258 after glitch is consistent density in the pulsar magnetosphere, e.g., when modeling with theresults here. the eclipse of the double pulsar system, giant pulses of the FromFigure6andFigure7,thepulsarwillgoup-right Crab pulsar, and magnetar X-ray spectra (Lyutikov 2008 in the P −P˙ diagram in the wind braking dominated case. and references therein), optical/ultraviolet excess of X-ray Therefore, for pulsars in the up right corner of the P −P˙ dim isolated neutron stars (Tong et al. 2010, 2011). Mag- diagram (e.g., high magnetic field pulsars and magnetars), netohydrodynamicalsimulationsoftheCrabnebulaalsore- theirtruedipolemagneticfieldmaybemuchlowerthanthe quire a much higher electron flow from the magnetosphere characteristic magnetic field. For magnetar SGR 1806−20, (Buhler & Blandford 2014 and references therein). Besides, its characteristic magnetic field is about 5×1015G (Tong asuperGoldreich-Julian currenthasbeenproposedinase- etal.2013).Thisdipolemagneticfieldistoohightounder- riestheoreticalworksaboutthepairscreationsintheaccel- stand comfortably (Vigano et al. 2013). Using the physical eration gap (Timokhin & Arons, 2013; Zanotti et al. 2012; braking mechanism in this paper, the dipole magnetic field Hirotani 2006). Combined with the results in this paper, it of SGR 1806−20 will be much lower. In the late stage of a ispossiblethattherearehighdensityplasmas(e.g.,103-104 pulsar,itwillbentandgodownwardintheP−P˙ diagram. timestheGoldreich-Juliandensity)inbothopenandclosed The Crab pulsar and other pulsars will not evolve into the field line regions in thepulsar magnetosphere. clusterofmagnetars.Furthermore,formagnetars,theyalso willgodownwardintheP−P˙ diagramwhentheyareaged enough. This may corresponds to the case of low magnetic field magnetars (Tong & Xu 2012). 3.1 Applications to other sources Here the updated pulsar wind model is employed to com- pare with the observations of the Crab pulsar. It can also beappliedtoothersources.Foreightpulsarswithmeaning- 4 CONCLUSIONS ful braking index measured (Espinoza et al. 2011; Lyne et Anupdatedpulsarwindmodelwhichconsideredtheeffectof al. 2015), their magnetospheric parameters can also be cal- particledensityandpulsardeathisdeveloped.Itisemployed culated, including magnetic field, inclination angle, particle to calculate parameters and simulate the evolution of the density etc. But the higher second frequency derivatives of Crabpulsar.Thebrakingindexn<3isthecombinedeffect some pulsars (Hobbs et al. 2010) may be attributed to the of magnetic dipole radiation and particle wind. The lower fluctuation of magnetosphere (Contopoulos 2007). During brakingindicesmeasuredbetweenglitchesarecausedbythe this process, more information will be helpful to constrain increasingparticledensity.Byaddingtheglitchparameters themodelparameters,e.g.,inclinationangle,secondbraking index. Their evolution on the P −P˙ diagram can be calcu- and a change of period derivative caused by the change of particledensity,thetheoreticalevolutioncurveiswellfitted lated, which will be similar to Figure 6. The pulsar PSR with observations (Figure 3). This may be viewed as glitch J1734−3333 has braking index n = 0.9±0.2 (Espinoza et inducedmagnetosphericactivitiesinnormalpulsars.Giving al. 2011). Its rotational energy loss rate will be dominated the model dependent acceleration potential drop, the mag- bytheparticlewind.AlsofromFigure6,thebrakingindices neticfield,inclinationangleandparticledensityoftheCrab of young pulsars will naturally be divided into two groups. pulsarcanbecalculated.Theevolutionofbrakingindexand Braking indices of pulsars in the first group are close to the Crab pulsar in P −P˙ diagram can be obtained. In the three,theirevolutiondirectionsaredowntotherightinthe P −P˙ diagram (P˙ ∝P2−n ∼P−1). Braking indices in the P−P˙ diagram,theCrabpulsarwillevolvetowardsthedeath valley, not to the cluster of magnetars (Figure 6). Different second groupareclose tooneandtheirevolution directions are up to the right (P˙ ∝ P2−n ∼ P1). Generally speaking, acceleration models arealso considered. Thepossible appli- cation of the present model to other sources (pulsars with sources in the second group will be older than the ones in braking index measured, and the magnetar population) is thefirstgroup.Atpresent,therearesomehintsfortheevo- also mentioned. lutionofpulsarbrakingindex(Espinoza2013).Futuremore observations will deepen ourunderstandingon this aspect. Glitch is a phenomenon of the pulsar suddenly spin- ningupandthenslowing downbyalarger rate.Inthepul- ACKNOWLEDGMENTS sar wind model, thelower brakingindex 2.3 duringglitches (of the Crab pulsar) is due to the larger out-flow particle Theauthorswould liketothanktheReferee BarsukovD.P. density.The increasing particle densityis a possible perfor- forhelpfulcommentsandR.X.Xu,J.B.Wangfordiscussions. mance associated with glitches. Glitches induced magneto- H.Tong is supported Xinjiang Bairen project, West Light spheric activities are very common in magnetars (Kaspi et Foundation of CAS (LHXZ201201), Qing Cu Hui of CAS, al.2003;Dib&Kaspi2014).Forthehighmagneticfieldpul- and 973 Program (2015CB857100). sar PSRJ1846−0258, it also shows asmaller brakingindex afterglitch(Livingstoneetal.2011).Thecauseofasmaller brakingindexmaybesimilartotheCrabpulsarcase.Dueto alarger particleoutflow,thepulsarshould experiencesome APPENDIX A: CONSIDERATION OF PULSAR net spindown after glitch. This has already been observed “DEATH” (Livingstoneetal.2010).SomeprimarycalculationsforPSR J1846−0258hasbeendonepreviously(Tong2014,basedon The treatment of Contopoulos & Spitkovsky (2006) is em- awindbrakingmodeldesignedformagnetars).Thesmaller ployed in the following. Assuming that the open field line Rotational evolution of the Crab pulsar in the wind braking model 9 regionsrotateataangularvelocityΩ butnottheangu- APPENDIX B: ASYMPTOTIC BEHAVIROS IN open lar velocity of theneutron star7. FIGURE 4 AND 5 Ω =Ω−Ω (A1) Inthepulsarwindmodel, thepulsarisbrakeddownbythe open death combinationofmagneticdipoleradiationandparticlewind. The particle density in this acceleration gap is κ times the Theeffectofparticlewindismoreandmoreimportantwith Goldreich-Julian charge density: pulsarspinningdown.AsshowninFigure4,theperiodand Ω B period derivative evolve up sharply with age. If the pulsar ρ ≈κ open , (A2) e 2πc brakingisdominatedbymagneticdipoleradiation,equation (1) can be written as: The energy taken away by theout-flowing particles is: ν˙ =−kν3. (B1) E˙ =2πr2cρ ∆φ. (A3) p p e the evolution of spin frequency is ν = (2kt)−1/2 (assuming Theperpendicularcomponentandparallelcomponentofthe initial spin frequency is very large). Correspondingly, the magnetic moment are respectively related to the magnetic periodandperiodderivativeevolutionwithtimearerespec- dipole radiation and particle acceleration. The rotation en- tively: P = (2kt)1/2, and P˙ = k(2kt)−1/2. If the pulsar is ergy loss rate can be written as: braked down mainly by particle wind, the braking index is 2µ2Ω4 Ω ∆φ about 1. The spin-down power law is : E˙ = E˙ +E˙ = (sin2α+3κ(1− death) cos2α) d p 3c3 Ω ∆Φ ν˙ =−kν. (B2) 2µ2Ω4 = η, (A4) the evolution of spin frequency is ν ∝ exp(−kt). Corre- 3c3 spondingly,theperiod and period derivativeevolution with with theexpression of η time are respectively: P ∝ exp(kt), and P˙ ∝ exp(kt). Ω ∆φ Clearly, the period and period derivative evolve faster by η=sin2α+3κ(1− death) cos2α. (A5) theexponential form. Ω ∆Φ For a constant acceleration potential ∆φ=3×1012V, correspondingly, REFERENCES Ω η=sin2α+54κ(1− death)R3B−1Ω−2cos2α. (A6) Abrahams A.M., Shapiro S.L., 1991, ApJ, 374, 652 Ω 6 12 Arons J., Scharlemann E. 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