The Crab Pulsar: Origin of the Crab Nebula’s Radio Pairs

Previously, we constructed a model—essentially a plausibility argument—in which the Crab Pulsar produces a spatially separated ion dominated and pair plasma dominated, magnetically striped relativistic wind, with the ion wind’s kinetic energy and electromagnetic Poynting fluxes being comparable. In this paper, the polar cap ion–photon pair production of that model is replaced with pair production by ion curvature synchrotron photons. The first primary ion curvature photons, and, contrary to conventional wisdom, also the first primary electron curvature photons, do not immediately convert into pairs. The primary beam particles continue to accelerate, and the actual photons that convert into pairs, which then short out the parallel electric field and terminate the acceleration, are produced by the further accelerated, higher energy particles. Simple estimates of the ensuing pair production cascade give pair multiplicities—the number of pairs per primary beam particle—of M± ≈ 6–8 × 104, comparable to standard calculations, but much less than the 3 × 106 value deduced by Rees and Gunn in order to sustain the Crab Nebula’s N± ≈ 1051 radio-emitting pairs against adiabatic expansion energy losses. Using a simple spin-down evolution model for the pulsar’s rotation frequency, the time-integrated pair cascade production driven by the primary ion beam can produce the N± ≈ 1051 radio pairs, whereas the primary electron beam produces about an order of magnitude fewer pairs.


Introduction
Many years ago, Arons (1983) suggested that the Crab Pulsar might emit an energetically significant flux of ions. Gallant & Arons (1994) subsequently showed that termination shocked ions might produce the Crab Nebula's wisps, and that the ion kinetic energy flux might transport a significant fraction of the Crab Pulsar's spin-down luminosity (Spitkovsky & Arons 2004). Recently, for the simple case of the orthogonal rotator, we presented a series of plausibility arguments for an ion dominated Crab Pulsar relativistic wind in which a super Goldreich & Julian (1969) flux of ions (assumed to be hydrogen) are strongly accelerated through a thin polar cap sheath (Coroniti 2017, hereafter CP). The accelerating parallel electric field was shorted out by the onset of ion-photon pair production, with the ions reaching final Lorentz factors of γ i ≈5-10×10 7 . A set of general magnetic geometry relativistic wind equations was derived, and the ion (and electron) field-aligned current was determined by the wind's Alfvén point condition to be about eight times the Goldreich-Julian value. The asymptotic ratio of the Crab wind's Poynting flux to ion kinetic energy flux was found to be s » ¥ 1 2 w . Thus, the ion dominated wind might resolve the so-called σproblem (Arons 2009), and provide wind parameters that are roughly consistent with the Gallant & Arons (1994) model of the wisps.
The objectives of this paper are two-fold. First, we replace the ion-photon pair production process of CP by the conversion of photons produced by ion curvature synchrotron radiation, the same process that dominates pair production from the primary electron beam in fast pulsars (Muslimov & Harding 1997;Hibschman & Arons 2001). Ion curvature photons pair produce, and therefore short out the accelerating parallel electric field, at a lower altitude than was found in CP for the ion-photon pair production process, which results in a somewhat lower final ion Lorentz factor. Thus, the ion curvature process is likely to dominate over the ion-photon process; however, a detailed computation that included the acceleration and pair production of both ion curvature and ionphoton processes would be needed to determine the fraction that each process contributed to the total production of pairs (and is well beyond the scope of this paper and the competence of the author).
We analyze the initial pair production for both the primary ion and electron curvature photons, finding that their pair formation fronts (PFF's) occur, not at the location where the first above pair energy threshold curvature photon is emitted, but at a distance along the field line that exceeds the local acceleration scale length ( ) in the exponentially accelerating sheath of CP. Thus, after the first curvature photon is emitted, the primary beam particles gain additional energy, and continue to radiate higher energy curvature photons until the local optical depth to pair production rises to about unity on a spatial scale much less than l a . At the PFF, the ion (electron) Lorentz factor γ i (γ e ) reaches a final value of about γ i ≈ 3.5×10 7 (γ e ≈7.5×10 7 ). A revised set of relativistic wind parameters is given, with the asymptotic ratio of Poynting to ion kinetic energy fluxes s » transition to optical at frequency ν 2 =10 14 Hz is L rad = 1.2×10 37 erg s −1 (Baldwin 1971;Baars 1972). The spectral luminosity (erg s −1 Hz −1 ) over that frequency range varies as n µ n a -L with a spectral index of approximately a @ 1 4. The radio-emitting pairs have a power-law energy distribution function g g µ a -+ ( ) ( ) f 2 1 between γ 1 <γ< γ 2 where n = 1 g n / 3 2 B 1 2 , n g n = / 3 2 B 2 2 2 , and pn = eB m c 2 B e is the electron gyro-frequency; m e is the electron/positron mass, c is the speed of light, and e is the electronic charge. For the Crab Nebula's average magnetic field strength of B≈2×10 −4 Gauss, γ 1 =1.9×10 2 and γ 2 =3.4×10 5 . For α=1/4, the average energy of the radio pairs ò ò turns out to be the geometric mean g gg á ñ = =( ) 8 10 1 2 1 2 3 . In the delta function approximation, the synchrotron power per Hz emitted by a single electron/positron is where σ T is the Thompson cross-section, n g n y = 3 2 sin c B 2 is the characteristic synchrotron frequency, and ψ is the pitchangle. Since the radio emission is approximately uniform within the Crab Nebula, the total radio luminosity is readily obtained by integrating over the frequency range and the pair distribution function to obtain a a s p where N ± is the total number of radio-emitting electrons and positrons. Setting L s =L rad , we obtain N ± ≈10 51 . Since the synchrotron lifetime of the radio pairs exceeds the age of the Crab Nebula, the usual assumption is that during the evolution of the Nebula, at least N ± ≈10 51 pairs must have been injected by the Crab Pulsar and/or accelerated from electrons produced in the filaments. Rees & Gunn (1974) estimated the pair injection rate  N by balancing the energy loss from the adiabatic expansion of the Nebula and the energy input from the newly injected pairs where (˙) R R is the "spherical" radius (expansion speed) of the Nebula. The expansion timescale t = =Ŕ R 3 10 s exp 10 yields = -N 3 10 s 40 1 . A long-standing conundrum is that the Rees & Gunn (1974) estimate for the injection rate of radio pairs is about 300 times higher than the injection rate of the high-energy pairs needed to produce the present optical-X-ray -gamma-ray luminosities (Kennel & Coroniti 1984a, 1984bArons 2009 where B * =2μ/a 3 is the polar cap magnetic field strength, μ≈4×10 30 G cm 3 is the dipole magnetic moment, a≈10 6 cm is the neutron star radius, and θ L =(Ωa/c) 1/2 is the colatitude of the polar cap's last closed field line-the boundary of the open magnetic flux. MultiplyingṄ R by the typical pair multiplicity M ± ≈ 10 4 obtained in most analyses of the pair production cascade (Arons & Scharlemann 1979;Muslimov & Harding 1997;Hibschman & Arons 2001) yields the 10 38 s −1 injection rate of high-energy pairs. The Rees & Gunn (1974) estimate for the radio pair injection rate would apparently imply that the pair multiplicity should be M ± ≈ 3×10 6 , which considerably exceeds the multiplicities found by typical pair cascade calculations. We will argue that the Rees and Gunn radio pair injection rate = -N 3 10 s 40 1 and the implied multiplicity M ± ≈ 3×10 6 actually represent "historical averages," and not necessarily the current values. The values of  N and M ± that we will obtain for the modified CP model are slightly larger than those found in most pair production cascade calculations, but still well below the Rees and Gunn radio pair estimate. However, we find that a simple model for the temporal evolution of the Crab Pulsar's rotation frequency results in a time-integrated number of injected pairs that is able to account for the N ± ≈10 51 inferred to exist in the Nebula today, with pair production by the primary ion beam exceeding that of the primary electron beam by about an order of magnitude.
From a different perspective, Atoyan (1999) argued that the existence of the Crab Nebula's radio electrons (pairs) might imply that the Crab Pulsar's rotation frequency at birth was much higher (6-10 times today's value, rather than the conventional estimate of about 2); his argument is based on assuming that the radio electrons are relic, and an attempt to fit the radio-optical-X-ray spectrum of the Nebula by analyzing the energy losses due to adiabatic expansion and synchrotron radiation. However, recent MHD modeling of the Crab Nebula by Olmi et al. (2014Olmi et al. ( , 2016 found that the distribution of the radio electrons in the Nebula is inconsistent with their being a relic population.
Section 2 determines the height of the electron and ion PFFs and the final Lorentz factors of the primary beams, and states the new asymptotic relativistic ion dominated wind parameters. Section 3 presents estimates of the ion and electron pair multiplicities from a very simple pair cascade model, and states the parameters for the relativistic pair plasma wind. Section 4 develops the temporal evolution of the pair injection rate into the Crab Nebula, and obtains the total number of pairs by integrating the injection rate over the age of the Nebula. Throughout this paper, we maintain the philosophy of CP-to develop simple estimates in order to determine whether a relativistic ion dominated outflow model for the Crab Pulsar is plausible; clearly, far more sophisticated calculations will be required to raise the certainty level from being merely plausible. Finally, the stated and calculated values of all physical parameters are based on the physical conditions that exist in the Crab Pulsar and Crab Nebula.

Threshold for Pair Production
In CP, we found that the polar cap parallel electric field accelerates primary beam ions and electrons to Lorentz factor where g =1.57 10 , and the dimensionless height s is used in all subsequent formulae; with θ * , f * being the colatitude and longitude in the polar cap and J 1 (k 1 θ * ) being a Bessel function; 0.58 is the maximum value of J 1 (k 1 θ * ) that occurs at k 1 θ * =1.8 or θ * /θ C =0.47; the ion (electron) outflow region is π/2<f * <3π/2 ( 2); the plane containing the rotation angular frequency and the magnetic moment is at f * =0. In CP the polar cap was approximated as a circle with radius aθ C , where the above value of θ C was taken at f * =0. This assumption allowed a simple analytic solution for the polar cap electric potential in terms of Bessel functions, and the above value of k 1 comes from setting the electric potential to zero at the boundary of the polar cap ( q = ( ) ) J k 0 C 1 1 and taking the first zero of the Bessel function ( j 11 =3.83). The actual variation of the polar cap boundary with f * depends on the specific model of the magnetosphere (Bai & Spitkovsky 2010). For making estimates, we will take Δ=1 and θ * /θ C =0.47 as fiducial parameters.
The characteristic angular frequency for curvature synchrotron radiation is for the electron beam.

Production of Curvature Photons
The production rate of curvature photons is , 7 s f C where α f is the fine structure constant. Using Equation (5) and integrating to obtain  ( ) N s s , the number of curvature photons produced by distance s, we find that the start of pair production For the fiducial parameters, = = k s ks 0.14, 1.635 . At these heights, from Equation (5)

Pair Production Opacity
The opacity κ E for the magnetic conversion of photons into pairs is given by Erber (1966) is the Compton wavelength of the electron. A curvature photon that is emitted in the direction of the magnetic field will acquire a pitch-angle y r » D a s C after it travels a distance Ds. Since photon conversion is a super-exponential process, the conventional assumption is that conversion occurs when c » L ( ) 8 3ln with L » ln 20 (Arons & Scharlemann 1979). Substituting for w s from Equation (6) is that the curvature photons produced by the primary electron beam convert "on-thespot," and that the PFF essentially occurs at the location where the first above threshold curvature photon is emitted. We will show below that, for the rapid accelerating electric potential of CP, this assumption is not quite correct. However,  q  s 1 i C indicates that the first above threshold curvature photons produced by the primary ion beam do not convert locally, and may not convert at all. In order to determine the final Lorentz factor of the both the primary electron and ion beams we have to construct a more accurate model of the PFF.

The PFF
The complete equation for the production and conversion of curvature photons is The production of pairs N ± (s) is given by Substituting from Equation (12) and integrating Equation (13) from s 0 to s yields two terms (Hibschman & Arons 2001 , is actually determined by the conversion of the more copious higher energy photons that are emitted by the still accelerating primary beam ions and electrons after the emission of the first curvature photon. Substituting for g ( ) s (Equation (5)) and changing variables to 3.83 256 81 The integral over dx″ in Equation (17) can be converted into an incomplete gamma function, and evaluated in the asymptotic limit to obtain The curly bracket in Equation (19) shows the value for electrons (ions) of 2.25×10 15 (3.64×10 5 ), and ) 10 3 for electrons (ions).
The integrand in Equation (16) is dominated by a sharp spike that occurs just below the point ¢ = x x (the super-exponential behavior of magnetic pair production; Hibschman & Arons 2001), and the value of the integral is essentially independent of the lower limit. Setting locates the height (s e , s i ) at which the first pair is produced (the pair production front) and the start of the synchrotron cascade. The rapid development of a dense pair plasma then shorts out the parallel electric field and terminates the acceleration of the primary beams.
By numerical integration, for the fiducial parameters Δ=1 and θ * /θ C =0.47, the ion (electron) PFF occurs at The final Lorentz factors are γ i =3.07×10 7 and γ e =7.5×10 7 . Thus, the final Lorentz factors and the height of the PFF considerably exceed their respective values at the emission point of the first curvature photon. In order to estimate the range for γ i and γ e , we performed calculations at a fixed longitude for a range of latitudes, and at a fixed latitude for a range of longitudes. We found that (1) for 1.0, γ i was essentially constant, and γ e decreased from 7.5×10 7 at  = 1 to 5.6×10 7 at =0.2, and then increased to 10 8 at  = 0.1.

Asymptotic Relativistic Ion Wind Parameters
For completeness, we briefly state the revised parameters of the relativistic ion wind; for details on the wind equations and their solution the reader is referred to Kennel et al. (1983) and CP. (The parameters of the relativistic pair plasma wind depend on the pair multiplicity, which is estimated in Section 3, so the pair plasma wind will be discussed in that section.) As in CP, we assume that the relativistic ion wind originates from an outflow that is uniform across the polar cap. For the specific value of the initial ion Lorentz factor, we chose γ i = 3.5×10 7 , which is just above the calculated value for the fiducial parameters. In developing the solution of the wind equations, the ion Lorentz factor is scaled to the parallel electric potential as g = F = e m c f 5.68 10 For the new estimates, the ratio of the electromagnetic Poynting flux to the ion kinetic energy flux at the star (=ratio of the total electromagnetic luminosity L r to the total ion luminosity L i ) is (in the notation of CP) where g=4/3π is a polar cap areal factor that was retained in CP in case the actual polar cap either differs from circular geometry and/or in the angular location of the last closed field lines (Arons 2012). The field-aligned current parameter J 0 , which scales the current density to the Goldreich-Julian current η R c, is determined by the Alfvén point condition to be which is slightly below the value of J 0 =8 found in CP. For the simple model of the wind region that was constructed in CP, the asymptotic ratio of Poynting flux to ion kinetic energy flux-the standard pulsar wind σ-parameter-is where Ω w is the fraction of 2π steradians that is filled with field lines from one polar cap, and is taken to be unity in the above estimate. In CP, the higher initial ion Lorentz factor/larger factor f i resulted in a smaller value of E, which gave s = ¥ 0.5 w i . However, to the limited accuracy of all of these estimates, s ¥ w i should probably just be considered to be of order unity. (2001), after acceleration stops at the PFF, the primary ion and electron beams will continue to radiate curvature photons. Rewriting Equation (6) in terms of energy normalized to m e c 2 ( w = m c e 2 ) in order to conform with the Hibschman & Arons (2001) notation, the curvature photon dimensionless energy is

Following Hibschman & Arons
Upon reaching the pitch-angle  y » L ( )( ( ) ) B B sin 8 3 ln  (2001) find that the minimum photon energy to produce pairs is (in their notation)

Pair Multiplicity
After the PFF, the total energy emitted by the primary beams as curvature radiation is , so that E R e is most conveniently written as In Equation (29) and below, since we have not solved for the spatial extent s of the pair production cascade, which may occur over a significant fraction of the star's radius (Hibschman & Arons 2001), the factor + ( ) s ln 1 is retained for completeness, where s may be of order unity; in what follows, we will take the logarithm factor to be of order one. The energy of the primary beam ions is essentially unchanged by the radiation of curvature photons. With * r q = + ( )( ) a s 4 3 1 C 1 2 , the total radiated energy from the ion beam is The multiplicity  M -the number of pairs per primary beam particle-at the end of the cascade (the nth generation) is where, for the purpose of making estimates, we will assume the bracket term is of order unity. For * q q = 0.47 C and g =3.5 10 i 7 , the primary ions produce only one gyrosynchrotron generation of pairs. Substituting Equation (25) into (31), the ion pair multiplicity can be conveniently written as for * q q = 0.47; C in making estimates, we will take the logarithmic factor to be of order unity.
Thus, both the primary electron and ion beams produce comparable pair multiplicities; both fall short of the Rees & Gunn (1974)  values do not need to apply today, but should actually be interpreted as "historical mean" values. The fundamental number whose origin needs to be understood is the »  N 10 51 radio-emitting pairs that exist in the Nebula today.

Asymptotic Relativistic Pair Plasma Wind Parameters
For an initial electron beam Lorentz factor of g =7.5  (22) with f e ) is = É 6.22 10 3 , and the field-aligned current parameter is J 0 =6.94, slightly less than for the ion wind. Finally, for the asymptotic pair plasma wind, the ratio of the Poynting flux to pair plasma kinetic energy flux in the asymptotic wind is s =¥ 1.2 10 w p 3 , which is a factor of about seven times lower than the classic s » ¥ 10 4 estimate due to the higher final electron beam Lorentz factor; in CP, we found * g » 10 e 7 .

Simple Spin-down Model
A simple (the simplest?) model for the spin-down evolution of the Crab Pulsar assumes that the magnetic moment is constant in magnitude and in orientation relative to the spin axis; although probably not correct, the evolution of the dipole and higher order moments is essentially unknown. Following Lyne et al. (1993), the model assumes that the evolution of the angular rotation frequency is given by W = -W W d dt k 5 2 , where, for simplicity, the braking index has been approximated as 5/2, which is close to the braking index that is measured between glitches (Lyne et al. 2015). The braking index and W k are assumed to be a constant in time. The spin-down model yields the result

Goldreich-Julian Number
The classic Goldreich-Julian particle injection rate per polar cap for the aligned rotator is Substituting for W( ) t from Equation (34) and integrating over the age of the Crab Pulsar, we find that the total number of particles that have been injected at the Goldreich-Julian rate since the birth of the Crab Pulsar is =( ) N t 4.9 10 R 0 44 . If the field-aligned current is » J 7 0 times the Goldreich-Julian current, combining both polar caps yields the total primary beam particle number = =( ) N t J N 2 6 . 8 1 0 R Tot 0 0 45 . The inferred number of radio-emitting pairs =  N 10 51 would then imply a pair multiplicity of = M 1.5 10 5 , significantly below the Rees & Gunn (1974) estimate, but above the current multiplicity estimate in our model, and in most other models, as well (Bucciantini et al. 2011;Arons 2012;Harding 2013).

Temporal Evolution of Characteristic Energies
The primary beam Lorentz factors at the PFF (s s , where the + (−) sign holds for the electron (ion) polar cap acceleration region. Since At t=0, the increase of g i e , above their current values is slightly less than four. Using the scaling relations from Equation (37), the energy of the primary beam curvature photons (Equation (25)) scales as 0 0 0 0 8 and the minimum cascade photon energy for gyro-synchrotron pair production (Equation (27)) scales as

Ion Cascade Pair Production
At t=0, the increased energy of the primary ions results in higher energy curvature photons that produce three generations of gyro-synchrotron photons with energies where the latitude * q has been left as a variable, but will be evaluated at * q q = 0.47 C below. As the pulsar spins down, g ( ) t i decreases, and, from Equations (39) and (40), min 3 , we find that the third generation is no longer produced after a time t = » ( t t 0.08 60 3 0 3 yr). A similar calculation yields that the second generation is not produced after time t = = ( t t 0.83 618 2 0 2 yr). The ion pair multiplicity for the nth generation (Equation (33) where, since the spatial extent of the cascade has not been determined by our simple analysis, the + ( ) s ln 1 factor has been retained for completeness, and is assumed to be of order unity. The production rate of pairs during the nth generation is then where we have added the two ion-halves of the two polar caps. Integrating Equation (42) with Equation (34) yields the total number of low energy pairs produced from pulsar birth to the present time

. If correct, since N i
Tot exceeds the 10 51 radio-emitting pairs that are inferred to exist in the Crab Nebula today, the implication would be that significant fluxes of the radio pairs must have escaped over the lifetime of the Crab Nebula. Recent MHD simulations (Porth et al. 2014) find that the magnetic field in the outer Nebula becomes dissipative and highly turbulent, which might account for the implied losses. However, an alternate possibility is that during the early epoch, the physics of the pulsar magnetosphere may be considerably different from today's highly relativistic response to essentially vacuum external conditions, which favor the production of pairs. For example, if the outflowing wind were to be choked by the Nebula's debris, polar cap pair production might be inhibited or prevented. Perhaps future observations and modeling of SN 1987A might shed some light on the early phase of pulsarnebula evolution.
If pair production is stymied during the early ion third generation epoch, the total number of ion pairs produced by the second and first generation cascades would then be =Ń 5 10 i Tot 51 . Finally, as done above, the pair production rate is conventionally scaled to the Goldreich-Julian fluxṄ R for the aligned rotator. For moderately oblique inclination angle i of the magnetic moment to the rotation axis, the Goldreich-Julian flux is reduced by cosi; for the Crab Pulsar » i cos 1 2. (For the orthogonal rotator, the Goldreich-Julian flux is even smaller m q p = Ẇ N a e; R C 3 however, in CP, reasonable arguments were presented that the basic physics of the primary ion and electron beams in the orthogonal rotator should also occur at least out to the range of inclination angles »   i 45 60 thought to encompass the Crab Pulsar's inclination Harding et al. 2008.) Thus, reducing the aligned rotatorṄ R by » i cos 1 2 may better estimate the pair production by the primary ion beam to be =Ń 2.5 10 i Tot 51 , which should probably be viewed as a fairly generous estimate since we evaluated all of the * * q f ( , ) angular dependences at the maximal location.

Electron Cascade Pair Production
The curvature radiation from the initial primary electron beam produces three generations of gyro-synchrotron pairs, and the third generation continues until t = » ( t t 0.633 465 3 0 3 yr). From Equation (31), the electron pair multiplicity for the nth generation scales as The total number of pairs produced by the primary electron beam is The values of the two terms within the brackets aré9 .92 10 , 1.18 10 3 3 . The coefficient of the bracketed integral terms is 8 ×10 46 , so that the total electron cascade pair production is =Ń 4.4 10 e Tot 50 , which should be reduced by cosi to =Ń 2.2 10 e Tot 50 .

Conclusion
Although today's pair production multiplicity is below the classical Rees & Gunn (1974) estimate of 3×10 6 deduced from adiabatic expansion energy losses, a simple spin-down model for the Crab Pulsar's rotational angular frequency indicates that higher multiplicities occurred in the past, and that the primary ion (electron) beam could produce somewhat more (less) than the 10 51 radio-emitting pairs that are inferred to exist today in the Crab Nebula. Clearly, the rough model estimates developed here at best indicate that the origin of the Nebula's radio pairs could be the Crab Pulsar's primary ion and electron beams. Far more sophisticated pair production calculations will be needed in order to confirm this conclusion.

Discussion
In the same "spirit" as adopted in CP, we have used "backof-the envelope" calculations of the Crab Pulsar's primary ion and electron beam-emitted curvature synchrotron photons to determine the location of the PFFs, the final beam Lorentz factors, and to obtain a very rough estimate of the gyrosynchrotron cascade's pair production multiplicity. We also addressed the Crab Nebula's radio electron/positron conundrum that the present cascade pair production rate falls short of the Rees & Gunn (1974) estimate of the injection rate needed to balance the radio pair's adiabatic expansion energy loss rate. Using a very simple spin-down model of the Crab Pulsar's rotation frequency, we found that the cascade pair production rate was considerably higher in the past, and that the timeintegrated number of generated pairs could plausibly account for the Nebula's 10 51 radio-emitting electron/positron pairs that exist in the Nebula today. We also found that the cascade driven by the primary ion beam produced about a factor of 10 more pairs than did the electron-driven cascade. Thus, a tempting conclusion is that the existence of the 10 51 radioemitting pairs is evidence that the Crab Pulsar's relativistic wind is dominated by ions; however, this conclusion should undoubtedly await confirmation by more sophisticated pair production calculations.
We have not attempted to explain the radio frequency spectral luminosity of the Crab Nebula ( n µ 3 ). Thus, the final pair cascade Lorentz factors are comparable to the average Lorentz factor of the Crab Nebula's radio pairs ( g g g á ñ = =( ) 8 10 1 2 1 2 3 , with g 1 being uncertain since it is derived from the ionospheric cutoff frequency). Clearly, the actual distribution function of the radio pairs will depend on the microscopic details of the cascade pair production process, the interaction of the pair plasma wind with the ion wind and with the termination shock, and the possibly turbulent acceleration dynamics within the Nebula. In any case, the proposed origin of the radio pairs-the Crab Pulsar-is consistent with the 2D and 3D MHD simulation results of Olmi et al. (2014Olmi et al. ( , 2016 that the radio pairs are continuously injected into the Nebula, and are not a relic population. For the orthogonal and moderately oblique rotators, the relativistic wind in the vicinity of the rotational equator contains magnetic stripes (Michel 1971) with alternating azimuthal magnetic field directions; one polarity (the other polarity) contains the ion dominated (the pair plasma dominated) wind. The two stripes have different Lorentz factors ( * g g » , 1 0 i p 3 ), and therefore have slightly different radial flow speeds The ion stripe will overtake the pair plasma stripe in a time  b p » W ( )( ) t 1 , during which time the wind will travel a radial distance   = »´W r c t c 7.5 10 5 , far smaller than the distance to the termination shock (»´W c 2 10 9 ). Thus, the ion and electron magnetic stripes are very likely to undergo significant magnetic field annihilation, and the ion and pair plasmas to mix, long before the wind reaches the shock.
The frequently quoted (Arons 2012) argument by Lyubarsky & Kirk (2001) that time dilation implies that the stripes do not annihilate before reaching the termination shock is incorrect, being derived from the same pitfall that underlies introductory physics Special Relatively paradox problems-events in different frames are not simultaneous. In order to illustrate, and presumably strengthen, their argument, Lyubarsky & Kirk (2001) assume that, in the wind frame, the annihilation wave travels at light speed, the fastest possible destruction of the magnetic field. Suppose the annihilation wave initiates at a stripe's neutral sheet, and travels in the negative radial direction at light speed. Consider a stationary observer at rest with respect to the wind. Since the annihilation wave travels at the speed of light, the Einstein Law of Velocity Addition says that all observers must measure the wave to travel at light speed. The next stripe's neutral sheet, a distance p W c 2 upstream, travels toward the stationary observer at essentially light speed. Thus, the oncoming neutral sheet and the annihilation wave essentially meet at the middle of the stripe (p W c ), and the magnetic field annihilates in a time p W as measured by the stationary observer. This result is readily confirmed from the exact Lorentz transformation of the initial (annihilation wave starts as the neutral sheet passes the stationary observer) and final (annihilation wave meets oncoming neutral sheet as seen by the stationary observer) events. The analyzes of Coroniti (1990) and Michel (1994), which show that the opposite magnetic polarity stripes annihilate well-within the wind zone, may be flawed for other reasons (many assumptions were made in those calculations), but not for violating Special Relativity. Recent simulation results by Cerutti & Philippov (2017) confirm their basic conclusion that the magnetic field in the stripes will dissipate before the wind reaches the termination shock.
In conclusion, in CP and this paper, we have presented plausibility arguments that the Crab Pulsar produces a relativistic wind with comparable electromagnetic and ion kinetic energy fluxes. The far asymptotic physical parameters of the Crab Pulsar's wind depend on the extent to which the striped magnetic field annihilates, and on the extent to which the energetically dominant ions and the pair plasma from the electron polar region mix and combine to form a coherent and steady wind flow.
If energetic ions do dominate the particle outflow from the Crab Pulsar, the interaction of the wind with the Crab Nebula will likely be considerably different than if the wind were purely positronic. As noted long ago by Gallant & Arons (1994), the Larmor radius of an energetic ion in the Nebula's downstream magnetic field is comparable to the radial distance from the Crab Pulsar to the termination shock (»3 0 17 cm) and to the distance between the wisps; thus, the interior of the collisionless ion shock is visible on the plane of the sky. The energetic ion Larmor radius is also comparable to the width of the ±10°opening angle of the X-ray torus (Aschenback & Brinkmann 1975;Weisskopf et al. 2000); thus, the termination shock, which probably occurs partly within a magnetic neutral sheet, will not be one-dimensional, but will likely have a quite complex three-dimensional structure. Within the Crab Nebula, the ballistic motion of the energetic ions is likely to excite both short and long wavelength plasma instabilities. The resulting turbulence may foster the creation of the localized reconnecting neutral sheets that are thought by many to be responsible for the Crab Nebula's gamma-ray flares (Abdo et al. 2011;Tavani et al. 2011;Cerutti et al. 2014;Lyutikov et al. 2018). The Crab Pulsar