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.
Modeling ultrarelativistic streaming plasma instabilities under the quasistatic approximation
GAP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院等离子体与核聚变研究所）；CEA, DAM, DIF（法国原子能及替代能源委员会）；Université Paris-Saclay, CEA, LMCE（巴黎萨克雷大学）；Helmholtz-Institut Jena（耶拿亥姆霍兹研究所）；Laboratoire d’Optique Appliquée (LOA), CNRS, École polytechnique, ENSTA, Institut Polytechnique de Paris（巴黎理工学院应用光学实验室）；Max-Planck-Institut für Kernphysik（马克斯·普朗克核物理研究所）
Quasistatic modeling of ultrarelativistic beam-plasma instabilities
Universidade de Lisboa（里斯本大学）；École polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；CEA（法国原子能和替代能源委员会）；Université Paris-Saclay（巴黎萨克雷大学）；Helmholtz-Institut Jena（亥姆霍兹耶拿研究所）；Max-Planck-Institut für Kernphysik（马克斯·普朗克核物理研究所）
