Scalings for the whistler instability growth rate and corresponding real frequency for bi-Maxwellian particle distribution

Whistler instability plays a crucial role in generating whistler mode waves, which serve as the primary driver for accelerating and scattering energetic electrons in the radiation belt. Whistler instabilities are excited by the electron temperature anisotropy, where electron population exhibits a bi-Maxwellian velocity distribution. In this study, we focus on examining the growth rate and the corresponding real frequency of the whistler instability. By solving the linear kinetic dispersion equation, we investigate the dependence of the maximum growth rate and the peak frequency with 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 plasma beta, temperature anisotropy and abundance of hot electrons increase. The peak frequency increases approximately linearly with hot electron abundance, while it exhibits an inverse trend with increasing plasma beta increases. We further establish an analytic scaling models for both the maximum growth rate and the peak frequency of whistler instability as functions of plasma beta, temperature anisotropy and abundance of hot electrons, respectively. The proposed scaling models exhibit good consistency with numerical solutions in most parameter regimes, especially when plasma beta > 0.2 and hot electrons abundance > 0.05. Our analytic models can be easily used to help characterize the physical consequences of whistler instabilities in the large-scale modeling of Earth's magnetosphere.