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⊥/T∥>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 suprathermal 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/βα∥, 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.