A comparative study of the filamentation and two-stream instabilities in weakly relativistic counter-streaming plasmas

Abstract

Counter-streaming plasma structures are ubiquitous in astrophysical sources of non-thermal radiations. We discuss the dispersion properties and the stability of this non-thermal particle distribution, which is modeled on the basis of the relativistic Jüttner-Maxwell distribution function in the correct laboratory frame of reference. In this work, we aim to construct analytical solutions of the dispersion relations and investigate the properties of the growth rate of the filamentation and two-stream instabilities in an unmagnetized and homogeneous counter-propagating plasma. The Maxwell and the relativistic Vlasov equations are used to derive the covariant dispersion relations that are valid in any (conveniently chosen) reference frame. Aperiodic solutions (ℜ(ω)≃0) to the covariant dispersion relations of the growing modes (ℑ(ω)>0) are demonstrated with the aid of analytical calculations. The dependence of the growth rate on the normalized bulk velocity β0=V0/c and thermal parameter μ=mc2/KBT is shown in graphic illustrations. We found that for both kinds of instabilities, growth rates are decreased by increasing the temperature and decreasing the bulk velocity. Therefore, the electrons at sufficiently low temperatures and with relativistic streams are capable of increasing the range of unstable wave numbers and consequently prevent the instability to cease at small wave numbers. The results indicate that under the same condition and in contrast to the non-relativistic regime, the filamentation instability has the largest growth rate and the electrostatic two-stream instability is in the next place.