We investigate the growth rate of field-aligned whistler-mode waves in space plasmas by using a fully relativistic treatment, including a recently developed relativistic kappa-loss-cone (KLC) distribution and a fully relativistic growth rate formula. Numerical calculations are carried out for a direct comparison between the new KLC distribution and the current kappa distribution, respectively. It is found that, in the lower wave frequency (e.g. ω ≲ 0.1Ωe), the wave growth by the KLC distribution is generally higher than that by the kappa distribution, due to a larger fractional number of the resonant electrons ηrel (which controls the wave growth) for the KLC distribution; but is lower in the higher wave frequency. The growth rates, as expected, tend to increase with the thermal anisotropy A, and the peak wave growth increases more rapidly for the kappa distribution. The relativistic anisotropy Arel basically decreases as the thermal parameter θ2 increases; whereas the fractional number of the resonant electrons ηrel is found to be large in the case of θ2 ∼ 150 keV, and results in a large wave growth. This indicates that hot electrons with typical energies of hundreds of keV may play a dominant role on the instability of whistler-mode waves. The results above have applications to plasma wave instability in the outer radiation belts of the Earth, the Jovian inner magnetosphere and other astrophysical plasmas where relativistic electrons are present.
Modulational instability of electron plasma waves in a cylindrical wave guide