  In situ observations of the solar wind show a limited level of particle
temperature anisotropy with respect to the interplanetary magnetic field
direction. Kinetic electromagnetic instabilities are efficient to prevent the
excessive growth of the anisotropy of particle velocity distribution functions.
Among them, the firehose instabilities are often considered to prevent the
increase of the parallel temperature and hence to shape the velocity
distribution functions of electrons and protons in the solar wind. We present a
non-linear modeling of the parallel firehose instability, retaining a kinetic
description for both the electrons and protons. One-dimensional (1D) fully
kinetic Particle-In-Cell simulations using the Energy Conserving semi-implicit
method (ECsim) are performed to clarify the role of the electron temperature
anisotropy in the development of the parallel proton firehose instability. We
found that in the presence of an electron temperature anisotropy, such that the
temperature parallel to the background magnetic field is higher than the
temperature in the perpendicular direction, the onset of the parallel proton
firehose instability occurs earlier and its growth rate is faster. The enhanced
wave fluctuations contribute to the particle scattering reducing the
temperature anisotropy to a stable, nearly isotropic state. The simulation
results compare well with linear theory. A test case of 1D simulations at
oblique angles with respect to the magnetic field is also considered, as a
first step to study the cumulative effect of protons and electrons on the full
spectrum of instabilities.
