Growth Rates of the Upper-Hybrid Waves for Power-Law and Kappa Distributions with a Loss-Cone Anisotropy

Abstract

Fine structures of radio bursts play an important role in the diagnostics of the solar flare plasma. Among them the zebras, which are prevalently assumed to be generated by the double-plasma resonance instability, belong to the most important ones. In this paper we compute the growth rate of this instability for two types of the electron distribution: a) for the power-law distribution and b) for the kappa distribution, in both cases with the loss-cone type anisotropy. We find that the growth rate of the upper-hybrid waves for the power-law momentum distribution strongly depends on the pitch-angle boundary. The maximum growth rate is found for the pitch angle $\theta_{\mathrm{c}} \approx 50^{\circ}$. For small angles the growth rate profile is very flat and for high pitch angles the wave absorption occurs. Furthermore, analyzing the growth rate of the upper-hybrid waves for the kappa momentum distribution we find that a decrease of the characteristic momentum $p_{\kappa}$ shifts the maximum of the growth rate to lower values of the ratio of the electron-plasma and electron-cyclotron frequencies, and the frequency widths of the growth rate peaks are very broad. But if we consider the kappa distribution which is isotropic up to some large momentum $p_{\mathrm{m}}$ and anisotropic with loss-cone above this momentum then distinct peaks of the growth rate appear and thus distinct zebra stripes can be generated. It means that the restriction of small momenta for the anisotropic part of distributions is of principal importance for the zebra stripe generation. Finally, for the zebra stripes observed on 1 August 2010, the growth rates in dependence on the radio frequency are computed. It is shown that in this case the growth rate peaks are more distinct than in usually presented dependencies of growth rates on the ratio of the plasma and cyclotron frequencies.