Phase space eigenmodes for cyclotron waves and instabilities in ring-distributed plasmas

Collisionless electrostatic modes in a spatially homogeneous magnetized plasma are considered through linear Vlasov-Poisson analysis. The function known as Gordeyev's integral for Maxwellian velocity distributions is extended to loss-cone, or ring, distributions with representations as trigonometric integrals and hypergeometric functions. Eigenmodes of perpendicular-propagating cyclotron waves are shown to be helical waveforms in phase space. Eulerian phase-space simulation of unstable propagating modes reveal fine structures and particle trapping.