Excitation of Langmuir waves by the lower energy cutoff behavior of power-law electrons

Langmuir waves (LWs), which are believed to play a crucial role in the plasma emission of solar radio bursts, can be excited by streaming instability of energetic electron beams. However, solar hard X-ray observations imply that the energetic flare electrons usually have a power-law energy distribution with a lower energy cutoff. In this paper, we investigate LWs driven by the power-law electrons. The results show that power-law electrons with the steepness cutoff behavior can excite LWs effectively because of the population inversion distribution below the cutoff energy (E _c ). The growth rate of LWs increases with the steepness index (δ) and decreases with the power-law index (α). The wave number of the fastest growing LWs (kλ _D ), decreases with the characteristic velocity of the power-law electrons ( (v_{c}=sqrt{2E_{c}/m_{e}}) ) and increases with the thermal velocity of ambient electrons (v _T ). This can be helpful for us to understand better the physics of LWs and the dynamics of energetic electron beams in space and astrophysical plasmas.