Observation has clearly shown that natural space plasmas generally possess a pronounced non-Maxwellian high-energy tail distribution that can be well modeled by a kappa distribution. In this study we investigate the proton cyclotron wave instability driven by the temperature anisotropy （T⊥/TH 〉1） of suprathermal protons modeled with a typical kappa distribution in the magnetosheath. It is found that as in the case for a regular bi-Maxwellian, the supratherreal proton temperature anisotropy is subject to the threshold condition of this proton cyclotron instability and the instability threshold condition satisfies a general form T⊥/T｜｜ - 1 = S/β｜｜^α, with a very narrow range of the fitting parameters： 0.40 ≤ α ≤ 0.45, and a relatively sensitive variation 0.27 ≤ S ≤ 0.65, over 0.01 ≤β｜｜ 〈 10. Furthermore, the difference in threshold conditions between the kappa distribution and the bi-Maxwellian distribution is found to be small for a relatively strong growth but becomes relatively obvious for a weak wave growth. The results may provide a deeper insight into the physics of this instability threshold for the proton cyclotron waves.
The electromagnetic wave growth or damping depends basically on the number density and anisotropy of energetic particles as the resonant interaction takes place between the particles and waves in the magnetosphere. The variance of both the number density and anisotropy along the magnetic field line is evaluated systematically by modeling four typically prescribed distribution functions. It is shown that in the case of ＂the positive anisotropy＂ （namely, the perpendicular temperature T⊥ exceeds the parallel temperature T｜｜）, the number density of energetic electrons always decreases with the magnetic latitude for a regular increasing magnetic field and the maximum wave growth is therefore generally confined to the equator where the resonant energy is minimum, and the number density is the largest. However, the ＂loss-cone＂ anisotropy of the electrons with a ＂pancake＂ distribution or kappa distribution keeps invariant or nearly invariant, whereas the ＂temperature＂ anisotropy with a pure bi-Maxwellian distribution or Ashour-Abdalla and Kennel＇s distributions decreases with the magnetic latitude. The results may provide a useful approach to evaluating the number density and anisotropy of the energetic electrons at latitudes where the observation information is not available.
