Two-stream-like instability in dilute hot relativistic beams and astrophysical relativistic shocks

TL;DR

This paper investigates the linear growth of electrostatic instabilities in the precursor region of relativistic shocks, revealing how different modes dominate depending on shock parameters and implications for gamma-ray burst shocks.

Contribution

It identifies the fastest growing electrostatic modes in relativistic shocks and analyzes their dependence on shock Lorentz factor and plasma conditions.

Findings

Unstable electrostatic modes are present in all plasma types and shock parameters.

Two competing modes, parallel and oblique, dominate growth rates depending on shock Lorentz factor.

The dominant mode varies with the perpendicular spread of particle momenta, affecting shock behavior.

Abstract

Relativistic collisionless shocks are believed to be efficient particle accelerators. Nonlinear outcome of the interaction of accelerated particles that run ahead of the shock, the so-called "precursor", with the unperturbed plasma of the shock upstream, is thought to facilitate additional acceleration of these particles and to possibly modify the hydrodynamic structure of the shock. We explore here the linear growth of kinetic modes appearing in the precursor-upstream interaction in relativistic shocks propagating in non and weakly magnetized plasmas: electrostatic two-stream parallel mode and electrostatic oblique modes. These modes are of particular interest because they are the fastest growing modes known in this type of system. Using a simplified distribution function for a dilute ultra-relativistic beam that is relativistically hot in its own rest frame, yet has momenta that are narrowly collimated in the frame of the cold upstream plasma into which it propagates, we identify the fastest growing mode in the full $k$-space and calculate its growth rate. We consider all types of plasma (pairs and ions-electrons) and beam (charged and charge-neutral). We find that unstable electrostatic modes are present in any type of plasma and for any shock parameters. We further find that two modes, one parallel ($k_\perp=0$) and the other one oblique ($k_\perp \sim k_\|$), are competing for dominance and that either one may dominate the growth rate in different regions of the phase space. The dominant mode is determined mostly by the perpendicular spread of the accelerated particle momenta in the upstream frame, which reflects the shock Lorentz factor. The parallel mode becomes more dominant in shocks with lower Lorentz factors (i.e., with larger momentum spreads). We briefly discuss possible implications of our results for external shocks in gamma-ray burst sources.