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 Aeff ≡ (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⊥/rL ≫ 1 is required (λ⊥ = 1/k⊥, where k⊥ is the wave vector perpendicular to the external magnetic field, and rL 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 Aeff 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.
Flexible simple-pole expansion of distribution functions