Nonthermal Particle Acceleration and Radiation

Although our dynamical model and synthetic images assume a thermal (essentially, a monoenergetic) pair distribution in the radiating plasma, they can be extended to the regime where the distribution is nonthermal, as is the case for the X-ray and optically emitting particles. The compressions in the field and plasma density associated with the shock structure should cause higher synchrotron emission for all electrons and positrons with Larmor radii smaller than the characteristic wisp separation, or for $\gamma_\pm < 10^{10}, E_\pm < 5 \times 10^{15}$ eV, emitting at $160 \; {\rm keV} < \varepsilon_{max} < 50-100$ MeV in the magnetic field whose strength increases from $\sim \sqrt{\sigma} B_{nebula} \approx 16 \mu$G at the leading edge of the shock structure to $\sim 300 \mu$G at and beyond the outer reaches of the X-ray torus (Kennel and Coroniti 1984a; de Jager and Harding 1992). Thus the wave motions seen in the Chandra band should display behavior similar to the optical wisps. That is, the features emitted from the X-ray inner ring discovered by Weisskopf et al. (2000) display similar variability as the optical moving wisps, and all propagate towards and through the X-ray torus. The observations do show such resemblances, at the qualitative level.

We have identified the Chandra X-ray ring as the site of the limit cycle in the unstable, reflected ion stream, not as the leading edge reverse shock in the pairs. The main reason to do this is the morphological similarities between the observed ring and our synthesized image. Whether the actual pair shock is located at the Chandra ring as concluded by Hester et al. (2002) or interior to it depends on the shock acceleration mechanism for the pairs. Below we consider the consequences of several shock acceleration schemes.

The pair shock heats the pair plasma to characteristic particle energies $\sim \gamma_{wind} m_\pm c^2 \approx 10^{12} \gamma_{w6.5}$ eV, where $\gamma_{w6.5} = \gamma_{wind}/10^{6.5}$. The downstream magnetic field at and immediately behind the pair shock is weak. For $\sigma \approx 3 \times 10^{-3}$, $B \approx 16 B_{s16}(r/R_{s\pm}) \mu$G. Pairs compressed and heated by the pair shock radiate in the optical and infrared, with characteristic frequency $\nu_c = 6 \times 10^{14} \gamma_{6.5}^2 (r/R_{s\pm})$ Hz. But, the synchrotron radiation time of newly shocked pairs at and just behind the pair shock is long, $T_{s\pm} \approx (3800/B_{s16}^2 \gamma_{6.5}) (R_{s\pm}/r)^2$ years, far larger than the local flow time $T_{\rm flow} = r/v(r) = 0.5 (R_{s\pm}/ 10^{17.2} {\rm cm})(r/R_{s\pm})^3$ years. Thus simple shock heating of the pairs implies optical emission does not start until $r > 10^{18}$ cm from the pulsar, which is larger than the radius of optical emission onset.

Gallant et al. (1992) demonstrated that the pair shock can indeed thermalize the upstream flow. The pair shock may, of course, also be a nonthermal particle accelerator, as has been assumed in all MHD models for the excitation of the Crab, starting with Rees and Gunn (1974). Diffusive Fermi Acceleration (DFA) in transverse relativistic shocks is often invoked as the mechanism for such acceleration (see Gallant 2002 for a review). Since the energy gains per encounter of a particle with the shock is $\Delta E \sim E$, this process might turn the pair shock into a nonthermal X-ray emitter, if the cross field diffusion time is short in the downstream medium, comparable to the cyclotron time (``Bohm diffusion'') - such diffusion must occur, if the requisite multiple encounters are to exist. Diffusion speeds exceeding the downstream flow speed ($\sim c/3$) require quite strong downstream turbulence, however (Bednarz and Ostrowski 1996), probably at levels higher than observed in the two-dimensional pair shock simulations of Gallant et al. (1992). In the absence of 3D simulations that allow evaluation of the turbulence level and which also allow for cross field diffusion, DFA at the pair shock is not excluded as the means of creating nonthermal emission, although the constraints on the required diffusion rates clearly are rather demanding.

Progressive acceleration with increasing radius, due to the resonant cyclotron absorption by the pairs of the high harmonic ion cyclotron waves emitted by the ions (Hoshino et al. 1992; Amato and Arons, in preparation) is an alternate to the commonly assumed DFA that has a natural setting in the context of the ion-doped relativistic shock whose spatial structure has been modeled here. The acceleration time is approximately the ion cyclotron time, which leads to energy gains such that X-ray emission from the Chandra ring requires placing the now unobserved pair shock at a radius between 2 and 3 times smaller than the radius of the X-ray ring. A full discussion of this distributed acceleration model will be given elsewhere.