This simple model can thus explain the polytrope law of index found in Section 3, if the electron velocity distribution can be approximated by a Kappa function of the form (10) with
This function is shown in Fig.3 and compared to its Maxwellian limit given in (11). Both distributions are rather similar for , i.e. at energies of the order or smaller than , whereas the Kappa function exhibits a supra-thermal tail at higher energies. Note that Eqs.(12)-( 13) yield with this value of . Since measured velocity distributions are often represented as the superposition of a cold and a hot Maxwellian, we compare in Fig.4 the above Kappa function with a sum of two Maxwellians having parameters of the order of those inferred from Voyager analysers in the range of Jovicentric distances explored here ([Sittler and Strobel 1987], [Bagenal 1994]).
Figure 3: Kappa distribution defined in Eq.(10) with given in (19), compared to its Maxwellian limit (11).
Figure 4: Kappa distribution defined in Eq.(10) with given in (19), compared to a distribution made of the sum of two Maxwellians (C,H) of densities and temperatures such that , (for this comparison, the cold temperature has been arbitrarily chosen equal to ).
In practice, the electron velocity distribution is not a priori expected to fit exactly such a Kappa function. Then, one will not find an exact polytrope law, but the density and temperature will still be anticorrelated along field lines, and mimic an approximate polytrope with . For example, the distribution made of a sum of two Maxwellians also results in a temperature increase with latitude; this can be easily understood: while the temperature of each Maxwellian does not change with z, the proportion of the hot component increases because it is less confined by the potential. However, with such a cold-plus-hot distribution having parameters of the order of those inferred from Voyager analysers (Fig.4), the temperature increase is rather small: Eq.(1) shows that a four-fold decrease in cold density (for example) produces an increase in by only 5% (instead of the factor of two observed here and explained by the Kappa distribution of Fig.4). Note also that the magnetic force, which should modify the above result since it does not derive from a conservative potential, is not expected to destroy the anticorrelation between density and temperature so long as the magnetic field variation is small over a characteristic scale length.