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.

Mon Feb 2 16:12:15 MET 1998