Modelling energetic particles by a relativistic kappa-loss-cone distribution function in plasmas

Energetic particles found in planetary magnetospheres and other plasmas, where mirror geometries occur, often exhibit two typical characteristics: a pronounced high energy tail and an anisotropy. A relativistic kappa-loss-cone (KLC) distribution function fκL is initially developed which incorporates features of the well-known kappa type and loss-cone type, i.e. the anisotropy behaves as a loss-cone distribution; the energy satisfies ∝ 1/v2(κ+1) for a relatively large velocity v as a kappa distribution fκ does and spreads ∝ 1/pκ+1 at the relativistic energies (where κ and p are the energy spectral index and the particle momentum, respectively). This indicates that the new distribution fκL obeys the power-law not only at the lower energies but also at the relativistic energies since the relativistic energy ∝ p. Numerical calculations are performed for a direct comparison between the new KLC distribution and the current kappa distribution, respectively. It is found that the regular kappa distribution generally decreases faster than the KLC distribution with the kinetic energy Ek especially when θ2 increases (where θ2 is the energy weight parameter), e.g. fκ/fκL ≲ 10−2 for Ek ≳ 2.0 MeV and θ2 ≳ 0.25. However, no big difference occurs between both distributions through energies up to ∼500 keV for θ2 ≲ 0.025. Furthermore, the regular kappa distribution containing either the temperature anisotropy or both the loss cone and temperature anisotropy is quite different from the KLC distribution. The new KLC distribution may be applicable to the outer radiation belts of the Earth, the inner Jovian magnetosphere and other plasmas (including the laboratory machine) where relativistic particles are present.