It is important to note that it is not necessary
to assume that the velocity distributions are Kappa functions
in order to derive such a density profile. One may adopt instead
the usual fluid description and solve the corresponding
equations along 
, assuming isotropic pressures and a
polytrope law of index 
. These equations read
where 
is the -aligned electric field. Here,
the electron and
ion (isotropic) pressures are
, and we consider the simplest case
where electrons and ions obey the same polytrope law
. Then 
, so that
One gets
Substituting in Eq.(26) and using (28)
with 
, one finds
Inserting the expression (3) of 
and integrating,
one
obtains
with H given in (25). This is equivalent to Eq.(
23) if 
and
are related by (18).
Hence a profile of the form (23) can be deduced
either
from Liouville's theorem - assuming Kappa distribution functions
(10) at z = 0, or from fluid equations - assuming
isotropic polytrope laws of index , for both electrons and
ions. Note that polytropes with exponent 
would give negative
densities farther than some finite distance z, which is
unphysical. This is not surprising since, as we have seen,
physical velocity distributions generally produce
anticorrelated density and temperature along z, which is not
compatible with polytropic exponents .
Finally, we note that the classical Gaussian density profile
corresponding to statistical equilibrium can
be recovered as a particular case of the above derivations,
either microscopic (Liouville) - by assuming a Maxwellian
velocity distribution at z = 0, or fluid - by taking the limit 
in the polytrope law. Indeed, Maxwellians of temperatures
are the limits for
of the Kappa
functions (10) and (20), for the electrons
and ions,
respectively. So that the density profile is obtained by taking
the same limit in Eq.(23),
i.e., referring to (11)
with the scale height given in (25). This is just
the result of the traditional isothermal torus models in the
particular case of one isotropic ion species ([Bagenal and Sullivan 1981]).
Similarly, in
the fluid description, this Gaussian profile can be
recovered by taking the limit in Eq.(
29).
