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tokens: is the proton/electron mass ratio, Θ = is the electron/proton parallel temperature ratio, and is the parallel electron beta parameter in the Maxwellian limit κ → ∞. The dispersion relation for bi-Maxwellian-distributed electrons is obtained from equation ( 14 ) by changing Z κ with the Maxwellian plasma dispersion function from equation ( 12 ). 3 EFHI SOLUTIONS We have solved the dispersion relation (equation 14 ) numerically, and analysed the unstable firehose solutions for a wide variety of plasma regimes susceptible to this instability. In this section, we present the main features of the EFHI, namely, growth rates, wave-frequencies, and wavenumbers, as well as the anisotropy thresholds, and restrict our discussions only to a number of representative cases. However, the anisotropy thresholds displayed and analysed in the second part of this section cover extended conditions encountered in space plasmas, for instance, in the solar wind and planetary magnetospheres. Since only the electrons are anisotropic, in the next analysis we omit the labelling subscript ‘e’, for the anisotropy, A , and the plasma beta parameter, β ∥ . The effects of Kappa-distributed electrons are triggered by their temperature anisotropy and the abundance of suprathermal populations, which is quantified by the finite (low) values of the power-index κ. 3.1 Unstable solutions First, we examine the growth rates and the wave-frequency of the EFHI for different plasma regimes conditioned in principal by the parallel plasma beta parameter β ∥ , the electron anisotropy A , and the power-index κ. The regimes identified in Figs 1 and 2 are specific to the firehose instability, when a magnetized plasma becomes penetrable by the LH EM fluctuations propagating parallel to the magnetic field with frequencies higher than the proton cyclotron frequency. All the unstable modes, i.e. with γ > 0 in Fig. 1 , exhibit this property that becomes evident in Fig. 2 , where their wavenumber dispersion extends to high frequencies exceeding Ω p . Increasing the presence of suprathermals, i.e. lowering the power-index κ, the range of unstable wavenumbers is restrained, but the wave-frequencies and the instability growth rates are enhanced. These effects are in general stimulated by increasing the plasma beta parameter β ∥ , the temperature anisotropy and the electron–proton temperature contrast Θ. Plots evidencing the influence of Θ are not shown here, being less relevant for the scope of this paper. The unstable solutions displayed in Figs 1 – 4 are obtained for the same value of this parameter, namely, for Θ = 4 in accordance to the observations in the slow solar wind (Newbury et al. 1998 ). Figure 1. Effects of the suprathermal electrons quantified by the power-index κ =2, 2.2, 3, 6, ∞, on the growth rates of EFHI for different plasma beta explicitly given in each panel. Here is the one used in equation ( 14 ). Figure 2. Effects of the suprathermal electrons (κ =2, 2.2, 3, 6, ∞) on the