Which One Grows Faster?

Many competing linear instabilities are likely to occur in astrophysical settings, and it is important to assess which one grows faster for a given situation. An analytical model including the main beam plasma instabilities is developed. The full 3D dielectric tensor is thus explained for a cold relativistic electron beam passing through a cold plasma, accounting for a guiding magnetic field, a return electronic current and moving protons. Considering any orientations of the wave vector allows to retrieve the most unstable mode for any parameters set. An unified description of the Filamentation (Weibel), Two-Stream, Buneman, Bell instabilities (and more) is thus provided, allowing for the exact determination of their hierarchy in terms of the system parameters. For relevance to both real situations and PIC simulations, the electron-to-proton mass ratio is treated as a parameter, and numerical calculations are conducted with two different values, namely 1/1836 and 1/100. In the system parameters phase space, the shape of the domains governed by each kind of instability is far from being trivial. For low density beams, the ultra-magnetized regime tends to be governed by either the Two-Stream or the Buneman instabilities. For beam densities equalling the plasma one, up to four kinds of modes are likely to play a role, depending of the beam Lorentz factor. In some regions of the system parameters phase space, the dominant mode may vary with the electron-to-proton mass ratio. Application is made to Solar Flares, Intergalactic Streams and Relativistic shocks physics. 1. INTRODUCTION Weibel, Filamentation, Two-Stream, Bell or Buneman instabilities are ubiquitous in astrophysics. They are involved in the physics of Solar Flares where relativistic electron beams are assumed to lose their energy through beam-plasma instabilities, producing hard X-ray emissions (Karlicky et al. as unstable particle streams through the intergalactic medium can magnetize an initially un-magnetized system. Such instabilities could also play an important role in explaining the origin of a variety of high energy photons sources including Supernova Remnants, some of these systems, it is assumed that cosmic rays are accelerated through shocks (relativistic or not) while the instability generated upstream by their interaction with the interstellar medium provides the magnetic turbulence eventually responsible for synchrotron radiation emissions Regardless of the context, the typical structure investigated consists in a beam-plasma system initially both charge and current neutralized. Initial charge neutralization implies the inclusion of positive and negative species, while current neutralization demands at least two streaming …