Plasma Instabilities of Streaming Cosmic Rays

A general expression is derived for the growth rate, due to particle-wave resonance, of plasma waves propagating in a spatially homogeneous, but anisotropic, relativistic plasma embedded in a uniform ambient magnetic field. This expression is good for plasma waves of arbitrary polarizations, wave vectors, and frequencies, provided that the growth rate is much less than the real part of the wave frequency. Applying this expression to an idealized model of cosmic rays streamIng parallel to the magnetic field in a cold background plasma, we find the system to be unstable against waves propagating at an arbitrary angle U to the field direction, provided that the phase velocity of the waves is less than the streaming velocity times the cosine of 0. If the streaming velocity is greater than the Alfvén velocity, then the maxi- mum growth rate occurs at frequencies much less than the cyclotron frequency of the cold ions. These low-frequency instabilities persist over a wide range of propagation angles since the growth rates vary only slowly with angle. These instabilities have e-folding times very much shorter than the lifetime of the cosmic rays in the galactic disk. An estimate of the time for the waves to gain a significant part of the bulk-streaming energy of the cosmic rays is obtained from the second-order equations. It is argued that the main effect of the waves is to reduce the mean streaming velocity by redistribution of momentum pitch angles with little transfer of energy