The electron-cyclotron maser （ECM） emission driven by nonthermal electrons is one of the most crucial mechanisms responsible for radio emissions in magnetized planets, for the interplanetary medium （IPM） and for the laboratory microwave generation devices. Major astrophysical observations demonstrate that nonthermal electrons frequently have a negative power-law spectrum with a lower energy cutoff and anisotropic distribution in the velocity space. In this paper, the effects of power-law spectrum behaviors of electrons on a ring-beam maser emission are considered. The results show that the growth rates of O1 and X2 modes decrease rapidly for small A （the dispersion of momentum u）. Because of the lower energy cutoff behavior, the nonthermal electrons with large a still can excite the ECM instability efficiently. The present analysis also includes the effects of parameter β （βu0 is the dispersion of perpendicular momentum ui, u0 the average value of u） on the instability. The growth rate of X2 mode decreases with parameter v0 （v0 = u⊥o/uo, U⊥0 is the average value of u⊥）. But for O1 mode, the relationship between the growth rate and v0 is complicated. It also shows that the growth rates are very sensitive to frequency ratio Ω （frequency ratio of electron cyclotron frequency to plasma frequency）.
The plasma temperature （or the kinetic pressure） anisotropy is an intrinsic characteristic of a collisionless magnetized plasma. In this paper, based on the two-fluid model, a dispersion equation of low-frequency （ω〈〈ωci, ωci the ion gyrofrequency） waves, including the plasma temperature anisotropy effect, is presented. We investigate the properties of low-frequency waves when the parallel temperature exceeds the perpendicular temperature, and especially their dependence on the propagation angle, pressure anisotropy, and energy closures. The results show that both the instable Alfven and slow modes are purely growing. The growth rate of the Alfven wave is not affected by the propagation angle or energy closures, while that of the slow wave depends sensitively on the propagation angle and energy closures as well as pressure anisotropy. The fast wave is always stable. We also show how to elaborate the symbolic calculation of the dispersion equation performed using Mathematica Notebook.
