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tokens: of equation 33 ). Finally, in Fig. 4 we show the temperature moment itself as given in equation ( 33 ). Figure 2. Kappa function f κ ( v , η(Θ, κ)) for isentropic electron flows for different values of kappa as a function of the particle velocity v in units of U 0 = 100 km s −1 . One can see that different transitions from a Maxwellian core to a power-law distribution are occurring along the heliosheath streamlines at an isentropic electron behaviour. Figure 3. The differential temperature moment of the electrons defined by the integrand of equation ( 33 ) is plotted as a function of the velocity v in units of U 0 = 100 km s −1 for different values of the parameter κ. This shows that for different permitted combinations of the parameters κ and Θ the contributions to the effective temperature are due to the higher electron velocities, the lower are the associated kappa values. Figure 4. The electron temperature, as a velocity-moment of the distribution function given by equation ( 33 ) as a function of the upper velocity border v ∞ in units of U 0 = 100 km s −1 , is shown as a function of the actual kappa value for isentropic electron flows. 4 NON-ISENTROPIC ELECTRON FLOWS AND THE ELECTRON ENTROPY In case that the electron flow along the streamlines, contrary to the assumption in the section before, develops non-isentropically due to the presence of energy sources or sinks, i.e. if the entropy of the electron fluid changes with the flow line element s , then one has to consider the following more complicate relation along streamlines: where d Q according to thermodynamical convention describes energy gains and losses per unit of the streamline element s in the co-moving volume Δ V , and T = T κ ( s ) is the local kappa temperature. Looking at equation ( 34 ), one can already qualitatively say that practically no change of the entropy has to be faced, if the temporal energy change compared to the given energy content Q is negligible, i.e. the latter condition for would lead to a quasi-isentropic behaviour. In order to check on this condition, we may now look into the pressure transport equation (equation 14 ) where one can identify two processes contributing to the change of the energy density Q ( s ) = ε( s ), namely: connected with the process of magnetic cooling (see Section 2 , equation 8 ) which is given by and the process of velocity-space diffusion (see Section 2 , equation 11 ) leading to the following term: Hence one can express the energy change per time by . Since we have already derived the solution for the local pressure P κ ( s ), one thus, based on that, can also obtain the entropy evolution with the following equation (see Section 2 , equation 14 ): where the actual kappa temperature is given by (see Heerikhuisen et al. 2008 ; Scherer et al. 2018 ) The kappa-entropy S κ ( s ) can then be calculated with the above differential equation, using the equation for P κ ( s ) (equation 14 ) derived further above in this paper in