Let us first consider the classical case where is a Maxwellian of temperature . One sees from (4) that the distribution remains Maxwellian for , with the same temperature, and, from (6), that all the moments vary with z as . This would justify the widely-used assumption of constant temperatures along magnetic field lines, if the particle velocity distributions were actually Maxwellian; of course, in this case, the density and temperature are not anticorrelated.
If the distribution is now a linear combination of Maxwellians, so that there is no more thermal equilibrium, the temperatures are no longer equal to each other, nor independent of z (although the temperature of each Maxwellian is independent of z). The generally increase with q, since higher-order moments favor components of higher temperatures. In particular, the effective temperature given by Eq.(1) is then . In the Appendix, we show analytically that with such a distribution and a monotonic potential which attracts particles to z = 0, all the generalized temperatures increase with z. Hence, since the density decreases with z, all the temperatures vary in anticorrelation with . An important consequence is that if a polytrope law does exist, its index is necessarily smaller than one (or just equal to one in the limiting case of a Maxwellian distribution.)
This generalizes to the temperatures the anticorrelation between density and temperature first shown by Scudder (1992a) in a general context, for the traditional temperature, using graphical arguments; this is a generic property of non-thermal distributions.