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Proceedingsofthe363.WE-HeraeusSeminaron:“NeutronStars and Pulsars”(Postersand contributedtalks) Physikzentrum BadHonnef, Germany,May.14−19,2006, eds.W.Becker,H.H.Huang, MPEReport291,pp.108-111 Polarisation of high-energy emission in a pulsar striped wind J. Pétri1 and J. Kirk Max-Planck-Institutfu¨r Kernphysik,Saupfercheckweg 1, 69117 Heidelberg - Germany 7 0 0 2 n Abstract. Recent observations of the polarisationof the An alternative site for the production of pulsed radi- a optical pulses from the Crab pulsar motivated detailed ation has been investigated (Kirk et al. 2002), based on J comparativestudiesoftheemissionpredictedbythepolar the idea of a striped pulsar wind, originally introduced 9 cap, the outer gap and the two-pole caustics models. by Coroniti (1990) and Michel (1994). Emission from the In this work, we study the polarisation properties striped wind originates outside the light cylinder and rel- 1 v of the synchrotron emission emanating from the striped ativistic beaming effects are responsible for the phase co- 0 wind model. We use an explicit asymptotic solution for herence of the synchrotron radiation. A strength of this 5 the large-scale field structure related to the oblique split modelis that the geometryofthe magnetic field,whichis 2 monopole and valid for the case of an ultra-relativistic the key property determining the polarisation properties 1 plasma.Thisiscombinedwithacrudemodelfortheemis- of the emission, is relatively well-known. 0 sivityofthe stripedwindandofthe magneticfieldwithin Inthiswork,weuseanexplicitasymptoticsolutionfor 7 0 thedissipatingstripesthemselves.Wecalculatethepolar- the large-scale field structure related to the oblique split / isation properties of the high-energy pulsed emission and monopole and valid for the case of an ultra-relativistic h p compareour resultswith opticalobservationsofthe Crab plasma (Bogovalov1999). This is combined with a crude - pulsar.Theresultingradiationislinearlypolarised.Inthe model for the emissivity of the striped wind and of the o off-pulse region, the electric vector lies in the direction of magnetic field within the dissipating stripes themselves. r t theprojectionontheskyoftherotationaxisofthepulsar, Wecalculatethepolarisationpropertiesofthehigh-energy s a in good agreement with the data. Other properties such pulsed emission and compare our results with optical ob- : as a reduced degree of polarisation and a characteristic servations of the Crab pulsar. v i sweep of the polarisation angle within the pulses are also X reproduced. 2. Stokes parameters r a Our magnetic field model is based on the asymptotic solution of the split monopole for the oblique rotator, valid for r r ,andmodified to take accountof afinite width 1. Introduction ≫ L of the current sheet. We add a small meridional compo- The high-energy, pulsed emission from rotating magne- nent in this sheet in order to prevent the magnetic field tised neutron stars is usually explained in the framework from becoming identically zero. The 3-dimensional geom- ofeither the polarcapor the outer gapmodels. Although etry of the currentsheet is shownin figure 1. In spherical the existence of such gaps is plausible (Pétri et al. 2002), polar coordinates (r,θ,ϕ) centered on the star and with these models still suffer from the lack of a self-consistent axis along the rotation axis, the radial field is small and solution for the pulsar magnetosphere. Nevertheless, re- neglected, Br ∼ BLrL2/r2, while the other components cent observations of the polarisation of the optical pulses are: from the Crab (Kanbach et al. 2003) motivated detailed r L B = B b η (∆ ,r,θ,ϕ,t) (1) comparative studies of the emission predicted by the po- θ L 1,2 θ θ r lar cap, the outer gap and the two-pole caustics mod- r L B = B η (∆ ,r,θ,ϕ,t) (2) els (Dyks et al. 2004). In all of these models, the radia- ϕ L r ϕ ϕ tion is produced within the light cylinder. However, the η (∆ ,r,θ,ϕ,t) = tanh[∆ (cosθ cosα+ (3) ϕ ϕ ϕ pulse profile is determined by the assumed geometry of r the magnetic field and the location of the gaps. None of sinθ sinα cos ϕ−Ω∗ t− v these models is able to fit the optical polarisationproper- 1 ∂η (∆ ,r,nθ,ϕ,t) (cid:16) (cid:17)o(cid:17)i ϕ θ η (∆ ,r,θ,ϕ,t) = (4) ties of the Crab pulsar. θ θ ∆ ∂ϕ θ J. Pétri & J. Kirk:Polarisation of high-energy emission in a pulsar striped wind 109 are given by the following integrals: Iω +∞ π 2π p+p+7/13 Qω (tobs) = s0(r,tret)cos(2χ˜)d3r(6)  Uω  Zr0 Z0Z0  sin(2χ˜)  where the retarded time is given byt = t +n r/c ret obs · and n is a unit vector along the line of sight from the pulsar to the observer.In this approximation the circular polarisation vanishes: V = 0. The function s is defined 0 by: p+3 p+1 2 B 2 s (r,t) = κK(r,t)D 1 ( n b)2 (7) 0 ωp−21 (cid:18)Γ − D · (cid:19) p whereω istheangularfrequencyoftheemittedradiation, andκisaconstantfactorthatdependsonlyonthenature of the radiating particles (charge q and mass m) and the Fig.1. 3D structure of the current sheet in the striped power law index p of their distribution: wind.Therotatingneutronstarislocatedattheoriginof the coordinate system. √3 1 3p+7 3p 1 κ = Γ Γ − Eu Eu 2π 4 12 12 (cid:18) (cid:19) (cid:18) (cid:19) p−1 Here, B is a fiducial magnetic field strength, v is the q 3 3 q 2 L | | | | (8) (radial) speed of the wind, Ω∗ = c/rL is the angular 4πε0mc (cid:18)m3c4(cid:19) velocity of the pulsar, with r the radius of the light L with Γ the Euler gamma function and the Doppler cylinder, α is the angle between the magnetic and rota- Eu D boosting factor =1/Γ(1 β n). The direction of the tion axes, b1,2 are parameters controlling the magnitude D − · localmagneticfieldintheobserver’sframeisgivenbythe of the meridional field in the two current sheets present unit vector b and the simplifying assumption has been in one wavelength, and ∆ are parameters quantifying θ,ϕ made that this field has no component in the direction of the sheet thickness. The functional form of B is moti- ϕ the plasma velocity: b β = 0. (In this case the magnetic vated by exactequilibria of the planar relativistic current fieldtransformationfro·mtherestframeB′totheobserver sheet (Kirk & Skjæraasen 2003). However, in these equi- frameB isjustB′ =B/Γand,thus,itsdirectionremains libriatheB component,whichhasanimportantinfluence θ unchanged.) The angle χ˜ measures the inclination of the on the polarisationsweeps, is arbitrary.The B we adopt θ local electric field with respect to the projection of the corresponds to a small circularly polarised component of pulsar’s rotation axis on the plane of the sky as seen in the pulsarwindwave,suchasisexpectedifthe sheetsare the observer’s frame. The degree of linear polarisation is formed by the migration of particles within the wave, as defined by described qualitatively by Michel (1971). Q2+U2 For the particle distribution, we adopt an isotropic Π= (9) electron/positron distribution given by p I Thecorrespondingpolarisationangle,definedasthe posi- N(E,p,r,t)=K(r,t)E−p (5) tionanglebetweenthe electricfieldvectorattheobserver and the projection of the pulsar’s rotation axis on the whereK(r,t)isrelatedtothenumberdensityofemitting plane of the sky is particles.Theradialmotionofthewindimposesanoverall 1/r2 dependence on this quantity, which is further mod- 1 U χ= arctan (10) ulated because the energization occurs primarily in the 2 Q (cid:18) (cid:19) currentsheet. The precise value ineachsheetis chosento fit the observedintensity ofeachsub-pulse. In addition,a 3. Results small dc component is added, giving the off-pulse intensity. For the emissivity, we use the standard expressions The upper left panel of Fig. 2 shows the intensity (Stokes for incoherent synchrotron radiation of ultra-relativistic parameter I) computed using our smoothed profile with particles. We assume the emission commences when the ∆ = 1, ∆ = 5, b = 0.1 and b = 0.08 for each sub- θ ϕ 1 2 wind crosses the surface r =r r . pulse. The electron density is 0 L ≫ The calculation of the Stokes parameters as measured intheobserverframeinvolvessimplyintegratingtheemis- 1 rLBL (p+1)/2 K = +ε 1 (11) sivity over the wind. For an observer, at time tobs, they r2(1−0.6ηθ) "(cid:18) rB (cid:19) − # 110 J. Pétri & J. Kirk:Polarisation of high-energy emission in a pulsar striped wind The degree of polarisation is shown in the two middle panelsofFig.2.Accordingtoourcomputations(left-hand panel) this displays a steady rise in the initial off-pulse phase,thatsteepensrapidlyasthefirstpulsearrives.Dur- ing the pulse phase itself, the polarisation shrinks down toabout10%.Theoreticallythemaximumpossibledegree of polarisationis closely relatedto the index p of the particle spectrum. In the most favorable case of a uniform magnetic field, it is given by p+1 Π = (12) max p+7/3 However, in the curved magnetic field lines of the wind, contributions of electrons from different regions have different polarisation angles. Consequently, they depolarise the overallresultwhensuperposed.We thereforeexpecta degreeofpolarisationthatisatmostΠ .Fortheexam- max 0 1 0 1 2 ple shown in figure 2, Π (p = 2) = 69.2%, well above max the computed value, which peaks at 52%. The lower panels of Fig. 2 show the polarisation an- Fig.2.Lightcurveofintensity,degreeofpolarisationand gle, measured against celestial North and increasing from positionangleofthepulsedsynchrotronemissionobtained North to East. Our model predicts this angle relative to for our model and measurements of these quantities for the projection of the rotation axis of the neutron star on theCrabpulsar.ModelswithLorentzfactorΓ=20(solid ◦ the sky, which we take to lie at a position angle of 124 , red) and 50 (dotted cyan) are shown. The bottom panels following the analysis of (Ng & Romani 2004). In the off- show the dependence on phase of the assumed magnetic pulse stage, the electric vector of our model predictions fieldcomponentsandtheparticledensityinthecomoving liesalmostexactlyinthisdirection,sinceitisfixedbythe frame. orientationofthedominanttoroidalcomponentB ofthe ϕ magnetic field. In the rising phase of the first pulse, B ϕ decreases, whereas B increases, causing the polarisation θ where the parameter ε =0.05 sets the minimum electron angle to rotate from its off pulse value by about 50◦, for density between the current sheets (in normalized units). the chosen parameters. However, for a weaker B contri- θ The denominator (1 0.6ηθ) introduces an asymmetry bution, as in the second pulse, the swing decreases. This − in the relative pulse peak intensity. The variation of the effectcanalsobecausedbyarelativelylargebeamingan- magnetic field and the particle density along the line of gle,(i.e.,lowLorentzfactorwind).Thebasicreasonisthat sight, are shown in the bottom panels of Fig. 2. contributionsfromparticleswellawayfromthesheetcen- The upper panels of Fig. 2 show the results of our ter are then mixed into the pulse, partially canceling the computations on the left and the corresponding observed contribution of the particles in the center of the current ◦ ◦ ◦ quantities(Kanbach et al. 2003)ontheright.Comparison sheet,whichfavorχ=124 90 ,andenforcingχ=124 . ± with the upper right-hand panel shows that the model Ontheotherhand,averyhighvalueoftheLorentzfactor reproduces the observed pulse profile quite accurately. or large values of b reduce the off-center contribution, 1,2 ◦ ◦ In this example, we adopt an obliquity α = 60 and leading, ultimately, to the maximum possible 90 sweep an inclination of the rotation axis to the line of sight between off-pulse (B -dominated) and center-pulse (B - ϕ θ ξ = arccos(n Ω∗/Ω∗ ) = 60◦ and set the radius at dominated) polarisations, followed by another 90◦ sweep · | | which emission switches on to be r = 30r . The posi- in the same sense when returning to the off-pulse. Thus, 0 L tion angle of the projection of the pulsar’s rotation axis in general, in the middle of each pulse, the polarisation ◦ is set to 124 (Ng & Romani 2004). The electron power angle is either nearly parallel to the projection of the ro- law index p = 2, as suggested by the relatively flat spec- tation axis, or nearly perpendicular to it, depending on trum displayed by the pulsed emission between optical the strength of B and on Γ. This interpretation is con- θ and gamma-ray frequencies (Kanbach 1998). Results are firmed by computations with Γ = 50 that show a larger shown for two values of the Lorentz factor of the wind: sweep, as shown in Fig. 2. Γ = 20 (solid line) and Γ = 50 (dotted line). For con- Theopticalpolarisationmeasurementssuggestthatin ◦ venience, the maximum intensity is normalized to unity. the centreofthepulsesthepositionangleiscloseto124 . Thetimescaleisexpressedintermsofpulsephase,0corre- In the declining phase of both pulses, the angle reaches spondingtotheinitialtimet=0and1toafullrevolution a maximum before returning to the off-pulse orientation. of the neutron star and thus to one period T∗ =2π/Ω∗. Note that in both cases the swing starts in the same di- J. Pétri & J. Kirk:Polarisation of high-energy emission in a pulsar striped wind 111 rection,(counterclockwiseinfigure 2). This is determined them onthe basisofobservations.However,the geometry by the rotational behavior of the B component, imply- of the magnetic field, which is the crucial factor deter- θ ing that this changes sign between adjacent sheets, as in mining the polarisation properties, is constrained in the Eq. (4). The observed off-pulse position angle is closely striped model to be close to that of the analytic asymp- alignedwiththeprojectionoftherotationaxisofthepul- totic solution of the split monopole. We have therefore sar, in accordance with the model predictions. presenteddetailed computations ofthe polarisationprop- Inadditiontomodels aimedatprovidinga framework erties of the pulses expected in this scenario. These pos- for the interpretation of the emission of the Crab pul- sessthecharacteristicproperty,uniqueamongstcurrently sar, we have performed severalcalculations with different discussed models, that the electric vector of the off-pulse Lorentz factors Γ, injection spectrum of relativistic elec- emission is aligned with the projection of the pulsar’s ro- trons p and inclinations of the line of sightξ. The general tation axis on the plane of the sky. This is in striking characteristics of the results are: For low Lorentz factors, agreementwithrecentobservationsoftheCrabpulsar.In independent of p and ξ, the relativistic beaming becomes addition the striped wind scenario naturally incorporates weaker and the pulsed emission is less pronounced, be- features of the phase-dependent properties of the polari- cause the observer receives radiation from almost the en- sation angle, degree of polarisationand intensity that are tire wind. For instance, taking Γ = 2 and p = 2 or 3, the also seen in the data.This underlines the need to develop average degree of polarisation does not exceed 20 % and the modelfurther,inorderto confronthigh-energyobser- the swing in the polarisation angle is less than 30◦. For vations of the Crab and other pulsars. In particular, the high Lorentz factors Γ 50, the strong beaming effect manner in which magnetic energy is released into parti- ≥ means that the observer sees only a small conical frac- cles in the current sheet remains poorly understood and tion of the wind. The width of the pulses is then closely the link between the asymptotic magnetic field structure related to the thickness of the transition layer. The de- and the pulsar magnetosphere is obscure. gree of linear polarisation flattens in the off-pulse emis- Acknowledgements. We thank Gottfried Kanbach for provid- sion while it shows a sharp increase followed by a steep inguswiththeOPTIMAdataandforhelpfuldiscussions.This decrease during the pulses. Due to the very strong beam- work was supported by a grant from the G.I.F., the German- ing effect, only a tiny part of the wind directed along the Israeli Foundation for ScientificResearch and Development. line of sight will radiate towards the observer. In the off- pulse phase, the polarisation angle is then dictated solely by the B component “attached”to the line of sight,and References ϕ the degree of linear polarisation remains almost constant Bogovalov, S. V.1999, A&A,349, 1017 in time. For very high Lorentz factors, the behavior of Coroniti, F. V.1990, ApJ,349, 538 polarisation angle and degree remain similar to those of Dyks,J., Harding, A.K., & Rudak,B. 2004, ApJ, 606, 1125 Fig. 2, with perfect alignment betweenpolarisationdirec- Kanbach,G. 1998, Advancesin SpaceResearch, 21, 227 tion (electric vector) and the projection of the pulsar’s Kanbach, G., et al, 2003, Proceedings of the SPIE, Volume rotationaxisontheplaneoftheskyintheoff-pulsephase 4841, 82 and two consecutive polarisation angle sweeps of 90◦ in Kirk, J. G., & Skjæraasen, O. 2003, ApJ, 591, 366 the same sense during the off-pulse to center-pulse and Kirk,J.G.,Skjæraasen,O.,&Gallant,Y.A.2002,A&A,388, L29 center-pulse to off-pulse transitions. This mirrors the fact Michel, F. C. 1971, Comments on Astrophysics and Space that emission comes only from a narrow cone about the Physics, 3, 80 line of sight of half opening angle θ 1/Γ. ≈ —. 1994, ApJ, 431, 397 ForgivenvaluesofΓandξ,theparticlespectralindex Ng, C.-Y., & Romani, R.W. 2004, ApJ,601, 479 paffectsonlytheaveragedegreeofpolarisationdegreebut Pétri, J., Heyvaerts, J., & Bonazzola, S. 2002, A&A,384, 414 notthelightcurvenorthepolarisationangle.Forexample, Pétri, J., & Kirk J., ApJLetters, 2005, 627, L37 ◦ takingΓ=10andξ =60 ,aspectralindexofp=2leads toanaveragepolarisationofΠ˜ =19.2%whereasforp=3 it leads to Π˜ =30.8%. 4. Conclusions In the striped wind model, the high energy (infra-red to gamma-ray) emission of pulsars arises from outside the light cylinder. It provides an alternative to the more in- tensively studied gapmodels. They all contain essentially arbitraryassumptionsconcerningthe configurationofthe emissionregionand the distribution function of the emit- ting particles,renderingit difficultto distinguishbetween