Closed-form analytical model of the electron whistler and cyclotron maser instabilities in relativistic plasma with arbitrary energy anisotropy

Detailed properties of the cyclotron maser and whistler instabilities in a relativistic magnetized plasma are investigated for a particular choice of anisotropic distribution function F(p_⊥²,p_z) that permits an exact analytical reduction of the dispersion relation for arbitrary energy anisotropy. The analysis assumes electromagnetic wave propagation parallel to a uniform applied magnetic field B₀e^_z. Moreover, the particular equilibrium distribution function considered in the present analysis assumes that all electrons move on a surface with perpendicular momentum p_⊥=p^_⊥=const and are uniformly distributed in axial momentum from p_z=-p^_z=const to p_z=+p^_z=const (so-called ‘‘waterbag’’ distribution in p_z). This distribution function incorporates the effects of a finite momentum spread in the parallel direction. The resulting dispersion relation is solved numerically, and detailed properties of the cyclotron maser and whistler instabilities are determined over a wide range of energy anisotropy, normalized density ω_p²/ω_c², and electron energy.