Harmonic electron-cyclotron maser emissions driven by energetic electrons of the horseshoe distribution with application to solar radio spikes

Content. Electron-cyclotron maser emission (ECME) is the favored mechanism for solar radio spikes and has been investigated extensively since 1980s. Most studies relevant to employ a loss-cone-type distribution of energetic electrons, generating waves mainly in fundamental X/O mode (X1/O1), with ratio plasma oscillation frequency electron gyrofrequency (${\omega}_ {pe}/{\Omega}_{ce}$) lower than 1. Despite great progress made this theory, one major problem how emissions pass through second-harmonic absorption layer corona escape. This generally known as escaping difficulty theory. Aims. We study harmonic generated by ECME driven electrons horseshoe solve spikes. Methods. performed fully kinetic electromagnetic PIC simulation ${\omega}_ {pe}/{\Omega}_{ce}$ = 0.1, corresponding strongly magnetized conditions flare region, characterized distribution. also varied density total ($n_e/n_0$) simulation. Results. obtain efficient amplification Z X2 modes, relatively weak growth O1 X3. With higher-density ratio, becomes more intense, rate energy conversion from into modes can reach $\sim$0.06% 0.17%, $n_e/n_0$= 5% 10%, respectively. Conclusions. find that horseshoe-driven lead an excitation X3 low value {pe}/{\Omega}_{ce}$, providing novel means resolving when applied The simultaneous be used explain some structures observed