aa r X i v : . [ a s t r o - ph . H E ] N ov The 11 th Asian-Pacific Regional IAU Meeting 2011NARIT Conference Series, Vol. 1, c (cid:13) Of Winds and Waves
John G. Kirk and Ioanna Arka Max-Planck-Institut f¨ur Kernphysik, Postfach 10 39 80, 69029 Heidelberg,Germany Institute de Planetologie et d’Astrophysique, University of Grenoble,Grenoble, FranceE-mail: [email protected]
Abstract.
Recent work on the properties of superluminal waves in pulsarwinds is summarized. It is speculated that these waves play an important rolein the termination shock that divides the wind from the surrounding nebula.
1. Introduction
In addition to pulses of radiation, rotation-powered pulsars are thought toemit a wind that powers the diffuse radiation or pulsar wind nebulæ (PWN)observed around many of them in the radio, optical, X-ray and gamma-ray bands.Synchrotron emission and inverse Compton scattering are the mechanisms mostlikely to be at work in PWN, and it has been evident for over half a century thatnot only the energy, but also both the relativistic electrons (or positrons) andthe magnetic fields in these nebulæ must be supplied by the central star[1].The basic idea is that the rotation of the neutron star couples to the windthrough the strong magnetic field anchored in its crust [2, 3]. As well aselectromagnetic fields oscillating at the rotation frequency of the pulsar, the windcontains a DC (phase-averaged) component of the magnetic field, electrons andpositrons created by cascades in the magnetosphere and, perhaps, a relativelysmall number of electrons and ions extracted from the stellar surface. As itpropagates away from the star, the ram pressure of the wind decreases, and,roughly where it equals the ambient pressure, a termination shock is formed.Rees & Gunn [3] simply assumed that the waves are absorbed at the terminationshock, leaving behind the energized particles and magnetic fields that fill thenebula. Kundt & Krotscheck [4], however, noted that electromagnetic wavescould also be reflected by the shock and build up inside it. This raises animportant question concerning the structure of the termination shock and thewind it encloses. Because the waves have large amplitude, they interact stronglywith each other and with the particle component. Rather than just a linearsuperposition of DC fields, particles, outward propagating waves and reflectedwaves, one must, therefore, look for self-consistent solutions, containing all ofthese, that match the outer boundary conditions.
2. The σ problem Estimates of the energy density in magnetic field at the light cylinder r L = c/ω (with ω the angular frequency of the pulsar) suggest that the wind is energetically Kirk & Arka dominated at launch by the waves and/or DC fields. On the other hand, outsidethe termination shock the particle pressure is comparable or larger than thepressure exerted by the fields (for reviews, see [5, 6]). The implied conversionof electromagnetic into kinetic energy is quite natural in many situations,for example, when an MHD outflow is collimated and accelerated into a jet.However, this kind of conversion does not appear possible in a pulsar wind,where the poloidal flux threads the neutron star [6]. In such a case, the MHDequations suggest the flow remains radial and its (supermagnetosonic) speedstays constant until the termination shock is reached.This, then, is the “ σ problem” (the ratio of Poynting to kinetic energy fluxis conventionally denoted by σ ): if the pulsar wind can be described by theequations of ideal MHD, there is no way in which a large value of σ near thestar can be reduced to unity or below before the termination shock is reached.Furthermore, if the termination shock is simply a discontinuity that obeys theusual jump conditions in an otherwise ideal MHD flow, the downstream plasmaremains magnetically dominated with large σ .
3. Dissipation in current sheets
To solve this problem, we obviously need to go beyond the ideal MHD description[7]. One possibility, suggested by Coroniti [8] and Michel [9], is to allow fordissipation in a current sheet embedded in a wind that is otherwise a cold,MHD flow. This is the “striped wind” picture. Conversion of the energy fluxfrom fields to particles by dissipation in the sheets causes the flow to accelerate[10, 11, 12] — in the simplest model the bulk Lorentz factor rises as Γ ∝ r / .Consequently, time-dilation reduces the effective dissipation rate seen in the lab.frame. It is possible to place upper and lower limits on the dissipation rate inorder to constrain the characteristic radius at which the oscillating componentsof the fields are annihilated. The results depend on the pair-loading of the flow,described by the parameter µ = L ˙ M c (1)where L is the luminosity and ˙ M the mass-loss rate. For the Crab, dissipationoccurs before the termination shock is reached only if µ < , a value somewhatlower than conventional estimates [13].Embedded current sheets also change our picture of the termination shock.On the basis of an analytical model and 1D PIC simulations, Lyubarsky &Liverts [14] and P´etri & Lyubarsky [15] suggested that an MHD shock sweepingthrough the cold portions of the striped wind would drive reconnection in theembedded current sheets and dissipate a substantial fraction of the magneticenergy, provided the resulting heated electrons were unmagnetized. For σ ≫ µ > a (2)where the strength parameter a of the incoming wave (striped wind) is the ratioof the quiver frequency of the electrons eB/mc to the angular frequency ω of the ulsar Wind Since a decreases as 1 /r in spherical geometry, (2) amounts toa lower limit on the radius at which such a process could operate: r > r crit (3) ≈ . × r L L / /µ where ( L / π ) × erg/s is the luminosity per unit solid angle carried by thewind.On the other hand, Sironi & Spitkovsky [18] recently considered drivenreconnection in the striped wind using 2D and 3D PIC simulations. Althoughtheir investigations were confined to the range 0 . < r/r crit <
4, they foundsubstantial dissipation in all cases. They also noted the appearance of high-energy particles accelerated by the first-order Fermi mechanism, and, on thebasis of a comparison of the nonthermal particle spectrum with the observedsynchrotron spectrum, suggested that, in the case of the Crab, the terminationshock should be located where r ≈ r crit /
3. If this interpretation is correct,the radius of the termination shock inferred from X-ray observations ( ∼ . µ ∼
50. This is much smaller than conventionally estimated, implying avery high pair loading and a mildly relativistic Γ ∼
10 outflow. An approximatelyspherically symmetric wind with these parameters would vastly overpopulatethe nebula with electrons and positrons. However, if we interpret the windparameters and, therefore, the radius of the termination shock, as latitudedependent, it cannot be ruled out that some part of the wind, for example theequatorial plane, contains a relatively dense, slow outflow of the type suggested.
4. Charge starvation
As well as giving rise to dissipation in current sheets, non (ideal-)MHD effectscan also manifest themselves via charge starvation, which can arise, for example,when the required currents demand relativistic motion of the current carriers[19, 20]. A relativistic two-fluid (electron-positron) approach can be used tomodel this effect. In the simplest case, with cold fluids and charge neutrality,both subluminal and superluminal nonlinear waves can be found [16].The subluminal wave resembles the striped wind, except that the current sheetis replaced by a static shear, with (cid:12)(cid:12)(cid:12) ~B (cid:12)(cid:12)(cid:12) constant (i.e., a circularly polarized wave).The phase profile of the shear can be chosen arbitrarily, but it is monochromaticin the simplest case of constant density. It is interesting to note that, as for thestriped wind, the non-MHD effects cause also this solution to accelerate withradius, in this case with Γ ∝ r , despite the fact that no dissipation process isinvolved [19]. However, at least for the monochromatic wave, acceleration startsrelatively far from the pulsar, where σ > a (4) There are several ways to define this important parameter. In the striped wind, where | B | isphase-independent and σ ≫
1, there is no ambiguity. For the superluminal waves discussed insection 5 we take a = eE max /mcω , where E max is the amplitude of the oscillating electric field.A covariant, gauge-independent definition for vacuum waves is given by Heinzl & Ilderton [17] Kirk & Arka u x , a n ω / ω p u x an Figure 1.
The radial four-velocity of the fluid, the strength parameter a , andthe refractive index n of circularly polarized superluminal waves close to thecut-off frequency for µ = 12, σ = 3.corresponding to r > Γ r crit (5)The solutions with superluminal phase speed resemble vacuum electromag-netic waves, in the sense that the displacement current plays a crucial role. Thefields are transverse, and satisfy (cid:12)(cid:12)(cid:12) ~E (cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12) ~B (cid:12)(cid:12)(cid:12) . The waves and particles are notlocked together in the way they are in the subluminal solutions, which makesthem more promising candidates for matching to a boundary condition at thetermination shock. There have been extensive investigations of these waves inthe literature (see [21] and references therein). They propagate only when thedensity falls below a critical value. In spherical geometry, the corresponding con-dition on the radius coincides with (3). Because of this, they cannot be launchedclose to the pulsar, and can only be realized if a striped-wind-like (subluminal)solution converts into a superluminal one either spontaneously, or because it isforced to do so by the boundary condition imposed by the termination shock.In this case, the conversion process and subsequent radial evolution and damp-ing of the mode should more properly be regarded as an integral part of the“termination shock” structure. ulsar Wind n log(R) b=0b=.2b=.65b=.95 Figure 2.
The refractive indices of linearly polarized superluminal waves closeto the cut-off frequency for µ = 10100, σ = 100 as a function of radius normalizedto r crit (see Eq. 3), for different values of the phase-averaged magnetic field
5. Superluminal waves
A first step towards understanding the role of superluminal waves is to determinethe propagation characteristics of those waves to which a given striped (or othersubluminal) wind can convert. This amounts to solving the two-fluid analogueof the shock jump conditions. Here we present a brief summary of some recentwork on this topic — a full description can be found in [21].Circularly polarized modes with zero phase-averaged field are the simplestto analyze, since the jump conditions can be solved analytically [16]. Figure 1illustrates the properties of these waves for parameters µ = 12, σ = 3 (and,therefore, Γ = 3). Although not in the range expected in pulsar winds, thisparameter set reveals the mode structure near the cut-off frequency particularlyclearly. The figure shows that for each value of the frequency ω in the lab.frame, measured in units of the proper plasma frequency ω p , two solutions existthat carry the prescribed particle, energy and momentum fluxes: a strong wavethrough which the plasma streams relatively slowly (dashed line) and a weakerone (solid line) through which the plasma streams rapidly ( u x is the four velocityalong the propagation direction). Transition to the the weak wave involvesalmost complete annihilation of the incoming field energy, leading to u x ≈ µ .The refractive indices of these modes coincide and correspond to superluminalphase speed n <
1. Because of the finite wave amplitude, the lab. frame frequency ω exceeds the proper plasma frequency ω p , which equals the cut-off frequencyfor linear waves. Kirk & Arka l og ( < u x > ) log(R) b=0b=.2b=.65b=.95 Figure 3.
The phase-averaged radial four-speed of the electron and positronfluids in the linearly polarized superluminal waves shown in Fig. 2.Linearly polarized waves are much more complicated, requiring a numericalsolution to the jump conditions. Nevertheless, the properties illustrated in Fig. 1apply also to these modes. Two solutions are available for each set of “upstream”parameters that characterize the corresponding subluminal solution. This isshown in Fig. 2, which differs from Fig. 1 in two ways. Firstly, we show therefractive index not as a function of frequency, but of radius in the pulsar wind,normalized to r crit given in Eq. (3). In this representation, the refractive indicesof the two solutions no longer coincide, since the proper plasma frequenciesassociated with each solution, differ although the lab. frame densities do not.Secondly, we show solutions for four different values of the phase-averagedmagnetic field, given by the parameter b = D ~B E / (cid:10) B (cid:11) / . It can be seenthat stronger DC fields push the cut-off radius further out from the pulsar.Furthermore, stronger fields display solutions with negative refractive index,corresponding to phase velocities directed radially inwards. Figure 3 shows thephase-averaged four-velocity of the plasma stream in the radial direction. Notethat the weak wave has h u x i ≈ µ only for zero averaged field. For finite valuesof the DC field, only a fraction of the incoming Poynting flux is carried by theoscillating fields and can be annihilated.
6. Summary
Because they are powerful enough to evacuate a relatively large cavity, pulsarwinds provide an environment in which electromagnetic fields dominate the ulsar Wind ∼ µ , can be regarded as magnetized, since their relativisticLarmor radius, µmc /eB (which in this case equals the effective inertial length c/ω p ) is smaller than the length scale c/ω . Here the superluminal modes cannotpropagate, because the plasma is sufficiently dense to screen out electromagneticwaves reflected from the termination shock. In this region, 2D and 3D PICsimulations indicate that dissipation could be driven by a standing shock front[18]. However, in the case of isolated pulsars, r crit lies well inside the point ofpressure balance with the external medium, making it doubtful that such a shockcould be supported.Outside r crit energized particles are unmagnetized. Driven reconnection atthe termination shock may still occur [15], but superluminal waves may alsoplay an important role. In section 5 we have briefly summarized some recentwork [21] on the properties of these waves for parameters thought appropriatein pulsar winds, and sketched their possible role. Although the physics of thesetwo scenarios remains to be investigated in detail, it seems likely that therewill be observable implications. Indeed, current sheets have been proposed asthe source of both the pulsed high-energy emission [22, 23] as well as the therecently observed flares in very high energy gamma-rays from the Crab [24, 25]. References [1] Piddington J H 1957
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