Unstable cyclotron waves in relativistic plasmas

Linearized equations are set up to describe cyclotron waves in a relativistic plasma immersed in a magnetic field, and in the presence of a non-relativistic plasma. Unstable waves are found to exist in the following cases: (i) when the momentum distribution is Maxwellian but anisotropic; (ii) when the momentum distribution is of the cosmic ray type, and is anisotropic with a lower momentum cutoff; (iii) when the momentum distribution is of type (ii) but isotropic and also possesses a loss cone. Maximum amplification rates are found for these three cases and are illustrated numerically by assuming physical values appropriate to the Crab Nebula and Cassiopeia A.