Instabilities of relativistic counterstreaming proton beams in the presence of a thermal electron background

A linear stability analysis is performed for two counterstreaming proton beams in the presence of a thermal electron background. Growth rates and polarization properties of unstable modes are calculated for various density ratios of the proton beams. It is found that in most cases, two unstable modes grow simultaneously: an electromagnetic filamentary mode that propagates perpendicular to the beam and an electrostatic mode that propagates parallel to the beam. The growth rates of the two modes are comparable, so that one expects that the instability would result in the development of a filamentary structure with a superimposed electrostatic pattern.