Instead of a superposition of a several Maxwellians, let us consider the simpler non-thermal distribution:
This generalized Lorentzian function is very convenient to model observed velocity distributions ([Vasyliunas 1968]), since it is quasi-Maxwellian at low and thermal energies, while its non-thermal tail decreases as a power-law at high energies, as generally observed in space plasmas; this is in line with the fact that particles of higher energy have larger free paths, and are thus less likely to achieve partial equilibrium. A generating process for such distributions has been suggested recently ([Collier 1993]). For typical space plasmas, generally lies in the range 2-6.
This ``Kappa" distribution tends to a Maxwellian for since
In this limit, all the temperatures . For finite , however, the temperatures are different and increase with q. In particular the traditional temperature is
and the effective temperature is
The larger , the closer the distribution is to a Maxwellian, and the closer the 's are to .
Substituting (10) into (4), one sees that the distribution at distance z is still a Kappa function having the same . In addition, as a consequence of the form (10), we have
Substituting this relationship into the integral (6) and changing variables to recover in the integrand, we get
(with , in order that the integrals converge). Since the density is the moment of order q = 0, this yields
Thus , and, since , we deduce
The 's and thus follow a polytrope law, which is independent of q. This generalizes the result of Scudder (1992a) to all the temperatures , and in particular to the temperature given by our measurement.
Hence with a Kappa distribution, the density and temperature obey a polytrope law, not only when the temperature is defined from the mean particle energy, but also when it is based on other moments of the distribution, a situation encountered with some measuring techniques.