Proceedingsofthe363.WE-HeraeusSeminaron:“NeutronStars and Pulsars”(Postersand contributedtalks) Physikzentrum BadHonnef, Germany,May.14−19,2006, eds.W.Becker,H.H.Huang, MPEReport291,pp.132-133 Electromagnetic Pulsar Spindown Ioannis Contopoulos Research Center for Astronomy,Academy of Athens,4 Soranou Efessiou Str., Athens11527, Greece 7 e-mail: [email protected] 0 0 2 n Abstract. We evaluate the result of the recent pioneer- tospheres of neutron stars requires solving for the self- a ing numericalsimulationsinSpitkovsky2006onthe spin- consistent behavior of plasma in strong fields, where field J down of an oblique relativistic magnetic dipole rotator. energy can dominate the energy of the plasma. This is 0 Our discussionis basedon our experience fromtwo ideal- difficult to do with the standard numerical methods of 1 ized cases, that of an aligned dipole rotator, and that of MHD which are forced to evolve plasma inertial terms 1 an oblique split-monopole rotator. We conclude that the even when they are small compared to the field terms. In v issue of electromagnetic pulsar spindown may not have these cases it is possible to reformulate the problem and 7 been resolved yet. insteadofsolvingfortheplasmadynamicsinstrongfields, 8 solve for the dynamic of fields in the presence of conduct- 2 1 ing plasma. This is the approach of force-free electrody- 0 namics”. In his recent pioneering work, Spitkovsky 2006 1. Introduction 7 managed to evolve numerically an oblique dipolar mag- 0 We believe that pulsars are spinning down magnetized netosphere for about 1.2 turns of the star, and claimed / h neutronstarswithnon-alignedrotationandmagneticaxes that the solution very quickly settles to a constant elec- p (oblique rotators). We also believe that pulsars lose rota- tromagnetic energy flux which depends on the magnetic - o tional energy through electromagnetic torques due to the inclination angle as astr ensettaobslpishhemreesn.tUonffloarrtguensactaelley,elaenctreixcaccutrreexnptrsesinsiothneifrormtahge- LSpitkovsky(θ)= B24rc63Ω4(1+sin2θ) . (2) electromagnetic (Poynting) flux energy loss L as a func- v: tion of the magnetic inclination angle θ remains still elu- Aswesaid,thisresultisveryimportant,andwhilewaiting i for independent numerical confirmation, we thought that X sive.Mostpeoplearecontentwiththeapproximationthat o we should be able to reproduce its main elements from pulsars spin down at the same rate as 90 vacuum dipole r first principles. We will, therefore, consider two idealized a rotators, namely that cases where we know the electromagnetic energy loss rate B2r6Ω4 analytically. L(θ)≈L (90o)≡ , (1) vac 6c3 2. The axisymmetric magnetic dipole rotator where,B,r,Ωarethepolarvalueofthemagneticfield,the radius of the neutronstar, and the star’s angular velocity Contopoulos,Kazanas&Fendt1999(hereafterCKF)first respectively. Knowing the exact electromagnetic energy obtainedthestructureofthe axisymmetricpulsarmagne- loss rate, observations of the pulse period P and period tospherewithadipolestellarmagneticfield.Theyshowed derivative P˙ would allow us to accurately infer the value that, at large distances from the central star, the asymp- of B. Moreover,the dependence of L on the magnetic in- totic structure is that of a magnetic split monopole, i.e. a clination angle θ has very important implications for the certain amount of initially dipolar magnetic flux evolutionanddistributionofpulsarsintheP−P˙ diagram πBr3Ω (Contopoulos & Spitkovsky 2006). Ψ =1.23 (3) open c In a realpulsar,the vacuumapproximationleading to eq.1isobviouslynotvalid,becausewithouttheestablish- stretches out radially to infinity in one hemisphere, and mentofmagnetosphericelectriccurrentsamagneticrota- returns to the star in the other hemisphere. Electromag- tor cannot generatepulses. The study of the oblique rela- netic spindown is due to the establishment of a poloidal tivistic magnetic rotator is, therefore, a formidable MHD electric current distribution problem as described clearly in Spitkovsky 2006: “Mod- ΩΨ Ψ I(Ψ)∼− 2− (4) eling of the structure of the highly magnetized magne- 4π (cid:18) Ψ (cid:19) open Ioannis Contopoulos: Electromagnetic Pulsar Spindown 133 which flows to large distances along open field lines and of the spindown luminosity). This calculation yields returns to the star through an equatorial current sheet πBr3Ω that joins at the light cylinder with the separatrix be- Ψ (θ)∼1.23 (1−0.2sin2θ) , (7) open c tween open and closed field lines. CKF obtained a first approximationforΨ andI(Ψ)selfconsistently,by re- which results in the following crude estimate of the spin- open quiring that the solution be continuous and smooth at down luminosiity, the light cylinder. The CKF solution has since been con- B2r6Ω4 firmed,improvedandgeneralizedinGruzinov2005a,Con- Lestimate(θ)∼ 4c3 (1−0.4sin2θ) . (8) topoulos 2005,Timokhin 2005,Komissarov2005,McKin- In particular, we see that L(90o) < L(0o), contrary to nery 2005 and Spitkovsky 2006. what both Gruzinov 2005b and Spitkovsky 2006 obtain. The electromagnetic spindown luminosity is thus ob- Note thatSpitkovsky’snumericalsimulationresultagrees tained as wellwithGruzinov’sestimate,andyetthelattermentions Ω Ψopen B2r6Ω4 Ω2 that he is not evensure about the sign of the θ term that L = I(Ψ)dΨ≈ ≈ Ψ2 CKF πcZ 4c3 6π2c open he obtains. Ψ=0 (5) We do not claim here that our above estimate has the (e.g. Beskin 1997). This result may be easily understood same weight as the result of a detailed numerical sim- as follows. As we said, the structure of the axisymmetric ulation. Interestingly enough, the recent discovery of an dipole rotator magnetosphere approaches asymptotically “on/off” pulsar (Kramer et al. 2006) offers some further that of an axisymmetric split-monopole (Michel 1991), insight from observations. The pulsar Kramer at al. dis- o and therefore, the spindown luminosity depends only on covered is thought to be a 90 oblique rotator with two the amount of open field lines Ψ . states of emission. During the “off” state, no pulses are open seen, and the neutron star is indirectly inferred to spin down at a certain rate L . During the “on” state, the off 3. The oblique magnetic split-monopole rotator pulsarisobservedtospindownatarateL =1.5L .It on off is very tempting to associate the “off” state with a vac- Bogovalov 1999 showed that eq. 5 is also valid for an uum state where the spindown rate is analytically known oblique split-monopololerotator.As longas currentsheet (eq. 1). The observations then imply that discontinuitiesarepresenttoguaranteemagneticfluxcon- servation, properties of cold MHD plasma flows do not L(90o)=L = 3L (90o)≡L ≡L(0o) . (9) on vac CKF depend on the direction of the magnetic field. In fact, all 2 properties obtained for the axisymmetric split-monopole In other words,a 90o oblique MHD rotatoris observedto rotator are the same for the oblique rotator as well. In spin down at the same rate as an aligned rotator. This particular, rotational losses of the oblique split-monopole result is more consistent with our crude estimate in eq. 8 rotatorareindependentofthe inclinationangle.Theyde- than with eq. 2. pend only on the square of the amount of open field lines We thus conclude that the discrepancy between the Ψ2open (obviously, the sign of Ψopen is irrelevant). two estimates is an indication that the issue of electro- Wemaynowderiveanimportantconclusionaboutthe magneticpulsarspindownmaynothavebeenresolvedyet, oblique dipole rotator which too, as we argued, becomes and that further investigation is needed, both analytical asymptoticallysplit-monopole-like:rotationallossesofthe and numerical. oblique rotatordepend indirectly on the inclination angle only through the amount of open field lines Ψ as open References Ω2 Beskin, V. S. 1997, Physics-Uspekhi,40, 659 L (θ)≈ Ψ2 (θ) . (6) estimate 6π2c open Bogovalov, S. 1999, A& A, 349, 1017 Contopoulos, I.,Kazanas, D.& Fendt,C.1999, ApJ,511, 351 (CKF) 4. The oblique magnetic dipole rotator Contopoulos, I.2005, A& A,442, 579 Contopoulos, I.& Spitkovsky,A. 2006, ApJ, 643, 1139 We have now come full circle. We may estimate the elec- Gruzinov, A. 2005a, Phys. Rev.Lett., 94, 021101 tromagnetic spindown luminosity of an oblique rotator Gruzinov, A. 2005b, astro-ph/0502554 through an estimate of the amount of open magnetic flux McKinney,J. C. 2006, MNRAS,368, L30 as a function of the inclination angle Ψ (θ). A crude Michel, F. C. 1991, Theory of Neutron Star Magnetospheres open estimate may be obtained if we calculate the amount of (Chicago: Univ.Chicago Press) Komissarov, S. S. 2006, MNRAS,367, 19 magnetic flux that crosses the light cylinder distance in Kramer et al. 2006, Science, 312, 549 the case of aninclined magnetostatic dipole at the origin, Spitkovsky,A.2006, ApJ,in press (astro-ph/0603147) and rescale our result by the CKF factor 1.23 (note that Timokhin, A. N., 2006, MNRAS,368, 1055 Gruzinov 2005b used a similar argument in his estimate