Growth rates of the electrostatic waves in radio zebra models

Zebras were observed not only in the solar radio emission but also in radio emissions of Jupiter and Crab Nebula pulsar. In their models, growth rates of the electrostatic waves play an important role. Considering the plasma composed from the thermal background plasma and hot and rare component with the Dory-Guest-Harris distribution, we compute the growth rates $\gamma$ and dispersion branches of the electrostatic waves in the $\omega - k_\perp$ domain. We show complexity of the electrostatic wave branches in the upper-hybrid band. In order to compare the results, which we obtained using the kinetic theory and Particle-in-cell (PIC) simulations, we define and compute the integrated growth rate $\Gamma$, where the "characteristic width" of dispersion branches was considered. We found a very good agreement between the integrated growth rates and those from PIC simulations. For maximal and minimal $\Gamma$ we showed locations of dispersion branches in the $\omega - k_\perp$ domain. We found that $\Gamma$ has a maximum when the dispersion branches not only cross the region with high growth rates $\gamma$, but when the dispersion branches in this region are sufficiently long and wide. We also mentioned the effects of changes in the background plasma and hot component temperatures.