On the kinetic instabilities of uniform magnetized plasmas with generalized loss-cone distribution functions

A general proof is given that in uniform magnetized plasmas described by generalized loss-cone distribution functions (loss-cone index l, thermal velocity α, and perpendicular spread α), electromagnetic, electrostatic, or coupled-mode instabilities are insensitive to the separate values of l and (α/α); they depend rather, on the effective thermal anisotropy A_eff ≡ (T/T)_eff-1, where (T/T)_eff ≡ (l + 1) (α²/α²). In the case of parallel propagation this statement is limited only by the linearization assumption; in the oblique propagation case, the additional condition λ/r_L 1 is required (λ = 1/k, where k is the wave vector perpendicular to the external magnetic field, and r_L is the Larmor radius). Thus, dispersion relations and their solutions obtained by using simple bi-Maxwellian distribution functions can be used directly for the complex case of generalized loss-cone distribution functions by simply replacing the anisotropy factor, A ≡α²/α²-1, by A_eff defined above. This result explains earlier conclusions that the growth rate of the whistler instability is independent of the explicit value of the loss-cone index l, for a given thermal anisotropy.