Long wavelength unstable modes in the far upstream of relativistic collisionless shocks

TL;DR

This paper analytically investigates long wavelength kinetic instabilities caused by relativistic particle beams in unmagnetized plasmas, revealing conditions for their growth or damping and implications for cosmic ray acceleration in relativistic shocks.

Contribution

It provides new analytic expressions for the growth rates and stability conditions of long wavelength modes in relativistic collisionless shocks, extending previous results and validating them with numerical simulations.

Findings

Long wavelength modes are Weibel-unstable for sufficiently collimated beams.

Modes become stable with large transverse velocity spreads.

Damping occurs in electron-proton shocks with moderate Lorentz factors.

Abstract

The growth rate of long wavelength kinetic instabilities arising due to the interaction of a collimated beam of relativistic particles and a cold unmagnetized plasma are calculated in the ultra relativistic limit. For sufficiently culminated beams, all long wave-length modes are shown to be Weibel-unstable, and a simple analytic expression for their growth rate is derived. For large transverse velocity spreads, these modes become stable. An analytic condition for stability is given. These analytic results, which generalize earlier ones given in the literature, are shown to be in agreement with numerical solutions of the dispersion equation and with the results of novel PIC simulations in which the electro-magnetic fields are restricted to a given k-mode. The results may describe the interaction of energetic cosmic rays, propagating into the far upstream of a relativistic collisionless shock, with a cold unmagnetized upstream. The long wavelength modes considered may be efficient in deflecting particles and could be important for diffusive shock acceleration. It is shown that while these modes grow in relativistic shocks propagating into electron-positron pair plasmas, they are damped in relativistic shocks propagating into electron-proton plasmas with moderate Lorenz factors \Gamma_{sh}\lesssim 100. If these modes dominate the deflection of energetic cosmic rays in electron-positron shocks, it is argued that particle acceleration is suppressed at shock frame energies that are larger than the downstream thermal energy by a factor greater than the shock Lorentz factor.