Steady State Electrostatic Layers from Weibel Instability in Relativistic Collisionless Shocks

It is generally accepted that magnetic fields generated in the nonlineardevelopment of the transverse Weibel instability provide effectivecollisionality in unmagnetized collisionless shocks. Recently, extensive twoand three dimensional simulations improved our understanding of the growth andsaturation of the instability in colliding plasma shells. However, thesteady-state structure of the shock wave transition layers remains poorlyunderstood. We use basic physical considerations and order-of-magnitudearguments to study the steady state structure in relativistic unmagnetizedcollisionless shocks in pair plasmas. The shock contains an electrostatic layerresulting from the formation of stationary, magnetically-focused currentfilaments. The filaments form where the cold upstream plasma and thecounterstreaming thermal plasma interpenetrate. The filaments are not entirelyneutral and strong electrostatic fields are present. Most of the downstreamparticles cannot cross this layer into the upstream because they are trapped bythe electrostatic field. We identify the critical location in the shocktransition layer where the electromagnetic field ceases to be static. At thislocation, the degree of charge separation in the filaments reaches a maximumvalue, the current inside the filaments comes close to the Alfv\'en limit, andthe phase space distribution function starts to isotropize. We argue that theradius of the current filaments upstream of the critical location is abouttwice the upstream plasma skin depth. Finally, we show that some downstreamparticles cross the electrostatic layer and run ahead of the shock into thepreshock medium without causing instability. These particles may play animportant role in particle acceleration.Comment: 9 pages, 3 figures, comments welcom