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Kappa distribution

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, tex2html_wrap_inline1663 generally lies in the range 2-6.

This ``Kappa" distribution tends to a Maxwellian for tex2html_wrap_inline2045 since


In this limit, all the temperatures tex2html_wrap_inline2047 . For finite tex2html_wrap_inline1663 , however, the temperatures tex2html_wrap_inline1969 are different and increase with q. In particular the traditional temperature is


and the effective temperature tex2html_wrap_inline1993 is


The larger tex2html_wrap_inline1663 , the closer the distribution is to a Maxwellian, and the closer the tex2html_wrap_inline1969 's are to tex2html_wrap_inline1979 .

Substituting (10) into (4), one sees that the distribution at distance z is still a Kappa function having the same tex2html_wrap_inline1663 . In addition, as a consequence of the form (10), we have


Substituting this relationship into the integral (6) and changing variables to recover tex2html_wrap_inline1921 in the integrand, we get


(with tex2html_wrap_inline2073 , in order that the integrals converge). Since the density is the moment of order q = 0, this yields





Thus tex2html_wrap_inline2081 , and, since tex2html_wrap_inline2083 , we deduce


The tex2html_wrap_inline1969 's and tex2html_wrap_inline1647 thus follow a polytrope law, which is independent of q. This generalizes the result of Scudder (1992a) to all the temperatures tex2html_wrap_inline1969 , and in particular to the temperature tex2html_wrap_inline1759 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.

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Michel Moncuquet
Mon Feb 2 16:12:15 MET 1998