Parametric scalings for the whistler instability growth by bi-Maxwellian energetic electron distributions

Whistler instabilities are a major generation mechanism for whistler-mode waves, which serve as one of the primary drivers for accelerating and scattering energetic electrons in radiation belts. These instabilities are excited by electron temperature anisotropy, which can be modeled using a bi-Maxwellian velocity distribution. In this study, we examined the growth rate and corresponding real frequency of the whistler instability. By solving the linear kinetic dispersion equation, we investigated the dependence of the maximum growth rate and peak frequency—defined as the real frequency at maximum linear growth—on plasma beta, temperature anisotropy, and hot electron abundance across a range of plasma conditions. Our results demonstrate that the maximum growth rate is significantly enhanced as the plasma beta, temperature anisotropy, and hot electron abundance increase. The peak frequency increases approximately linearly with hot electron abundance, whereas it exhibits an inverse dependence on plasma beta. We further established analytic scaling models for both the maximum growth rate and peak frequency of whistler instabilities as functions of plasma beta, temperature anisotropy, and abundance of hot electrons. The proposed scaling models exhibit good consistency with numerical solutions in most parameter regimes, especially for η_he ≥ 0.1 and β_||he > 0.5. Our analytic models can serve as practical tools for rapid parameter surveys and can be readily integrated into large-scale magnetospheric models.