Revisit of non-linear Landau damping for electrostatic instability driven by blazar-induced pair beams

Abstract

We revisit the effect of non-linear Landau damping on the electrostatic instability of blazar-induced pair beams, building on an earlier study of electrostatic-wave growth rates calculated for realistic pair-beam distributions. The new aspect in this paper is a simplified 2D model in ${\bf k}$-space that is used to study the evolution of the electric-field spectrum and to calculate the relaxation time of the beam. We verified that the 2D model is an adequate representation of the 3D physics. We find that non-linear Landau damping, once it operates efficiently, transports essentially the entire wave energy to small wavenumbers where wave driving is weak or absent. Formally, the relaxation time of the pair beam then is longer than the inverse Compton scattering time. We added collisions as a subdominant damping process for the waves and found that it reduces the wave intensity at very small $k$. Consequently, non-linear Landau damping is less efficient and the relaxation time of the pair beam reduced, albeit not as much as to be less than the inverse-Compton loss time. Any other loss process will act similarly, and a full description of the spectral evolution of the electrostatic waves is crucial for calculating the relaxation time of the pair beam.