  In the solar wind plasma an excess of kinetic temperature along the
background magnetic field stimulates proton firehose modes to grow if the
parallel plasma beta parameter is sufficiently high, i.e., $\beta_{p
\parallel}\gtrsim 1$. This instability can prevent the expansion-driven
anisotropy from increasing indefinitely, and explain the observations.
Moreover, such kinetic instabilities are expected to be even more effective in
the presence of suprathermal Kappa-distributed populations, which are
ubiquitous in the solar wind, are less affected by collisions than the core
population, but contribute with an additional free energy. In this work we use
both linear and extended quasi-linear (QL) frameworks to characterize the
unstable periodic proton firehose modes (propagating parallel to the magnetic
field) under the influence of suprathermal protons. Linear theory predicts a
systematic stimulation of the instability, suprathermals amplifying the growth
rates and decreasing the instability thresholds to lower anisotropies and lower
plasma betas ($\beta_{p \parallel}<1$). In perfect agreement with these
results, the QL approach reveals a significant enhancement of the resulting
electromagnetic fluctuations up to the saturation with a stronger back reaction
on protons, leading also to a faster and more efficient relaxation of the
temperature anisotropy.
