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The frequency distribution of solar wind protons, measured in the vicinity of Earth’s orbit, is customarily plotted in (β_∥,T_⊥/T_∥) phase space. Here,T_⊥/T_∥is the ratio of perpendicular and parallel temperatures, andβ_∥= 8πnT_∥/B²is the ratio of parallel thermal energy to background magnetic field energy, the so-called “parallel beta,” with ⊥ and ∥ denoting directions with respect to the ambient magnetic field. Such a frequency distribution, plotted as a two-dimensional histogram, forms a peculiar rhombic shape defined with an outer boundary in the said phase space. Past studies reveal that the threshold conditions for temperature anisotropy–driven plasma instability partially account for the boundary on the high-β_∥side. The low-β_∥side remains largely unexplained despite some efforts. Work by Vafin et al. recently showed that certain contours of collisional relaxation frequency,ν_pp, when parameterized byT_⊥/T_∥andβ_∥, could match the overall shape of the left-hand boundary, thus suggesting that the collisional relaxation process might be closely related to the formation of the left-hand boundary. The present paper extends the analysis by Vafin et al. and carries out the dynamical computation of the collisional relaxation process for an ensemble of initial proton states with varying degrees of anisotropic temperatures. The final states of the relaxed protons are shown to closely match the observed boundary to the left of the (β_∥,T_⊥/T_∥) phase space. When coupled with a similar set of calculations for the ensemble in the collective instability regime, it is found that the combined collisional/collective effects provide the baseline explanation for the observation.

The quasi-steady states of collisionless plasmas in space (e.g., in the solar wind and planetary environments) are governed by the interactions of charged particles with wave fluctuations. These interactions are responsible not only for the dissipation of plasma waves but also for their excitation. The present analysis focuses on two instabilities, mirror and electromagnetic ion cyclotron instabilities, associated with the same proton temperature anisotropyT_⊥>T_∥(where ⊥, ∥ are directions defined with respect to the local magnetic field vector). Theories relying on standard Maxwellian models fail to link these two instabilities (i.e., predicted thresholds) to the proton quasi-stable anisotropies measured in situ in a completely satisfactory manner. Here we revisit these instabilities by modeling protons with the generalized bi-Kappa (bi-κpower-law) distribution, and by a comparative analysis of a 2D hybrid simulation with the velocity-moment-based quasi-linear (QL) theory. It is shown that the two methods feature qualitative and, even to some extent, quantitative agreement. The reduced QL analysis based upon the assumption of a time-dependent bi-Kappa model thus becomes a valuable theoretical approach that can be incorporated into the present studies of solar wind dynamics.