The conversion of the Crab pulsar wind from one dominated by Poynting flux
close to the star to one dominated by particle-born energy at the termination
shock is considered. The idea put forward by Coroniti (1990) and criticised by
Lyubarsky & Kirk (2001) that reconnection in a striped wind is responsible, is
generalised to include faster prescriptions for the a priori unknown
dissipation rate. Strong acceleration of the wind is confirmed, and the higher
dissipation rates imply complete conversion of Poynting flux into particle-born
flux within the unshocked wind.Comment: 4 pages, to appear in "Young Neutron Stars and Their Environments"
  (IAU Symposium 218, ASP Conference Proceedings), eds F. Camilo and B. M.
  Gaensle

Kirk, J. G.

Skjaeraasen, O.

English

arXiv

The conversion of the Crab pulsar wind from one dominated by Poynting flux close to the star to one dominated by particle-borne energy at the termination shock is considered. The idea put forth by Coroniti (1990) and criticized by Lyubarsky and Kirk (2001) that reconnection in a striped wind is responsible, is generalized to include faster prescriptions for the a priori unknown dissipation rate. Strong acceleration of the wind is confirmed, and the higher dissipation rates imply complete conversion of Poynting flux into particle-borne flux within the unshocked wind

Kirk, J.

MPG.PuRe

The Sigma Problem of the Crab Pulsar Wind

We investigate how rapidly Poynting flux can be converted into kinetic energy in the wind of the Crab pulsar by implementing into a global MHD model explicit prescriptions for the dissipation rate. A lower limit is given by limiting the maximum drift speed of the current carriers to that of light an upper limit follows from demanding that the dissipation zone expand only subsonically in the comoving frame. A further prescription is obtained by assuming that the expansion speed is limited by the growth rate of the relativistic tearing mode. In each case solutions are presented which give the Lorentz factor of the wind containing a transverse oscillating magnetic field component as a function of radius. We find that the Poynting flux can be dissipated before the wind reaches the inner edge of the Nebula if the pulsar emits electron positron pairs at a rate greater than 1.E40 per secon

Skjæraasen, O.

Young Neutron Stars and Their EnvironmentsIAU Symposium, Vol. 218, 2004F. Camilo and B. M. Gaensler, eds.The σ problem of the Crab pulsar windJ.G. KirkMax-Planck-Institut fu¨r Kernphysik, Postfach 10 39 80,69029 Heidelberg, GermanyO. SkjæraasenObservatoire de Strasbourg, 11 rue de l’Universite´, F–67000 Strasbourg,FranceAbstract. The conversion of the Crab pulsar wind from one dominatedby Poynting flux close to the star to one dominated by particle-born en-ergy at the termination shock is considered. The idea put forward byCoroniti (1990) and criticised by Lyubarsky & Kirk (2001) that recon-nection in a striped wind is responsible, is generalised to include fasterprescriptions for the a priori unknown dissipation rate. Strong acceler-ation of the wind is confirmed, and the higher dissipation rates implycomplete conversion of Poynting flux into particle-born flux within theunshocked wind.1. IntroductionA puzzling feature of the relativistic MHD wind of the Crab pulsar is that itappears to arrive at its termination shock as a particle dominated flow (Rees& Gunn 1974; Kennel & Coroniti 1984; Emmering & Chevalier 1987). This isdifficult to understand, because the wind almost certainly carries its energy byPoynting flux close to the star, and, given that relativistic MHD winds collimateonly weakly (Beskin, Kuznetsova & Rafikov 1998; Chiueh, Li, & Begelman 1998;Bogovalov & Tsinganos 1999; Lyubarsky & Eichler 2001, Lyubarsky 2002) it isnot obvious how the transformation to particle-born flux can occur, or why itis apparently not observed. The acceleration intrinsic to axisymmetric force-free winds (Buckley 1977; Contopoulos & Kazanas 2002) does not alleviate thedifficulty, since the flow is still magnetically dominated outside the sonic pointwhere such solutions lose their validity.An obliquely rotating neutron star drives a wind which, near the equatorialplane, consists of alternating stripes of magnetic field of opposite polarity. Themagnetic field is linked alternately to the open field lines emerging from differentmagnetic hemispheres, as these are swept around the rotational equator. Thisled Coroniti (1990) to suggest that reconnection or, more accurately, dissipa-tion at the current sheet separating the alternating polarities, might effect therequired conversion. He estimated the dissipation rate by assuming the thick-ness of the current sheet was approximately equal to the gyro radius of the hotparticles it contained and found that the magnetic field could be annihilated12 J.G. Kirk & O. Skjæraasenbefore the wind reached the termination shock. However, Coroniti’s treatmentneglected the acceleration of the wind flow. Using the same micro-physics esti-mate for dissipation, Lyubarsky & Kirk (2001) showed that this effect results inan annihilation rate far too small to be effective.Kirk & Skjæraasen (2003) recently identified the dissipation estimate ofCoroniti as a lower limit and examined the effect of alternative micro-physicsprescriptions, including an upper limit on the dissipation rate. Their findings,which we paraphrase in this paper, re-open the possibility that the conversion ofPoynting flux to particle-born flux occurs through dissipation of magnetic fieldenergy at the current sheet of a striped wind.2. Small wavelength approximation and dissipation prescriptionsThe approach of Lyubarsky & Kirk (2001) and Kirk & Skjæraasen (2003) isbased on computing the slow evolution of an entropy wave as it is convectedoutwards by the stellar wind. The method employs a separation of length scalesinto a short one, identified with the wavelength of the magnetic field reversals,and a long one, identified with the radius, i.e., the scale on which the densitychanges as a result of the spherically expanding flow. The equations of conser-vation of particles and energy/momentum are then ordered in the small param-eter formed from the ratio of these lengths. The entropy wave whose evolutionis sought can be described in several ways. One is in terms of the parameterswhich describe the Harris sheath (Harris 1962; Hoh 1968); another is in termsof the parameters of two uniform fluid layers — one hot and unmagnetized, theother cold and magnetized — separated from each other by discontinuities indensity, velocity and magnetic field, but having equal pressures. Each of theseconfigurations satisfies the zeroth order conservation equations, indicating thatthe wave does not evolve on this short length-scale.On the long length scale, however, evolution of the wave is possible. Itis computed by regularising the expansion, i.e., by demanding that the secu-lar terms in the first-order equations vanish. The conservation equations bythemselves are not sufficient to solve for all the parameters of the wave as afunction of radius — an additional constraint is required, reflecting the need fornew physics input to determine the rate at which dissipation proceeds. In MHDsimulations, the new input needed to describe dissipation is a functional relationbetween the current and the electric field: an anomalous conductivity. For ourcomputations, however, it suffices to prescribe one of the wave parameters (thethickness of the Harris sheath, for example) in terms of the others.The Coroniti (1990) prescription implies a relative drift speed of the op-positely charged species roughly equal to that of light. In other words, thereexists no Harris equilibrium for a sheath thinner than that assumed by Coroniti.The thinner the sheet, the slower the dissipation, so that Coroniti’s prescriptionprovides a lower limit on the dissipation rate. Kirk & Skjæraasen (2003) con-sider two alternative prescriptions: that the sheath thickness is governed by the(linear) growth rate of the tearing instability, as suggested by Lyubarsky (1996),and that the expansion speed of the sheath is limited to its internal sound speed.The motivation for the latter choice stems from the observation that only the os-cillating component of the magnetic field in the entropy wave can be dissipated.APS Conf. Ser. Style 3Figure 1. The Crab pulsar wind with maximum rate of dissipation(sonic sheath expansion) and particle injection rate into the Nebula of3 × 1040 s−1. Shown as functions of r/rL, (where rL is the radius ofthe light cylinder) are the ratio ξ of the particle density in the currentsheet to that in the magnetized wind, the Lorentz factor γ in units ofits maximum value µ (= 2 × 104 in this case), the thickness ∆ of thesheath in units of pirL and the ratio ξ/∆. The termination shock islocated at log(r/rL) ≈ 9.This suggests that the boundaries of the hot fluid must be in causal contact inorder to guarantee that they advance into regions of opposite magnetic polarityat equal speeds. Thus, expansion at the internal sound speed provides an upperlimit to the dissipation rate.3. Implications for the Crab pulsar windThe dissipation of the magnetic field in the wind of the Crab pulsar depends onthe particle load carried by the wind. For a particle loss rate of N˙ < 6×1038 s−1,the theoretical upper limit on the dissipation rate lies below the lower limit givenby the Coroniti prescription, so that the treatment is inconsistent. In this case,slow evolution of the entropy wave is impossible. Presumably the system adjustsby either becoming unsteady or by steadily converting the flux into a mode4 J.G. Kirk & O. Skjæraasenwhich propagates in the fluid frame (e.g., an electromagnetic mode). For 6 ×1038 s−1 < N˙ < 3× 1040 s−1 none of the three dissipation prescriptions succeedsin converting significant amounts of Poynting flux before the termination shock isencountered. For N˙ > 3×1040 s−1, however, dissipation at the fastest permittedrate, as depicted in Fig. 1, results in complete conversion before the terminationshock is encountered. This rather high level of pair injection into the Nebulacannot be explained in terms of current theories of cascade formation in the innermagnetosphere (Hibschmann & Arons 2001), but is nevertheless consistent withthe observed number of radio-emitting electrons (Gallant et al 2002).Can the conversion of Poynting flux to particle-born flux be observed di-rectly? Arons (1979) noticed that conditions in the wind will make such a signalpulse at the rotation frequency of the star. The obvious implication that theoptical to gamma-ray pulses are indeed the signature of this conversion is fur-ther supported by the approximate correspondence of the sheet geometry to theobserved pulse form (Kirk, Skjæraasen, & Gallant 2002).Acknowledgments. We thank Yves Gallant and Yuri Lyubarsky for help-ful discussions.ReferencesArons J. 1979 Space Sci.Rev., 24, 437Beskin V.S., Kuznetsova I.V., & Rafikov R.R. 1998 MNRAS, 299, 341Bogovalov S.V., & Tsinganos K. 1999 MNRAS, 305, 211Buckley R. 1977 MNRAS, 180, 125Chiueh T., Li Z.Y., & Begelman M.C. 1998 ApJ, 505, 835Contopoulos I., & Kazanas D. 2002 ApJ, 566, 336Coroniti F.V. 1990 ApJ, 349, 538Emmering R.T., & Chevalier R.A. 1987 ApJ, 321, 334Gallant Y.A., van der Swaluw E., Kirk J.G., & Achterberg A. 2002 in “NeutronStars in Supernova Remnants”, Eds: P.O. Slane, B.M. Gaensler, ASPConference Series Vol. 271, 99Harris E.G. 1962 Nuovo Cimento, 23, 115Hibschman J.A., & Arons J. 2001 ApJ 560, 871Hoh F.C. 1968 Physics Fluids, 9, 277Kennel C.F., & Coroniti F.V. 1984 ApJ, 283, 694Kirk J.G., & Skjæraasen O. 2003 ApJ, 591, 366Kirk J.G., Skjæraasen O., & Gallant Y.A. 2002 A&A, 388, L29Lyubarsky Y. 1996 A&A, 311, 172Lyubarsky Y. 2002 MNRAS, 329, L34Lyubarsky Y., & Eichler D. 2001 ApJ, 562, 494Lyubarsky Y., & Kirk J.G. 2001 ApJ, 547, 437Rees, M. J., & Gunn, J. E. 1974, MNRAS, 167, 1

The "sigma" problem of the Crab pulsar wind

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