Let 
be the electron velocity distribution
at z = 0,
which we assume for simplicity to be isotropic. This
approximation is reasonable since the bulk of the electrons
has been inferred to be roughly isotropic in this region: an
observation made during a Voyager roll maneuver was found to be
compatible with 
at 
latitude
for the bulk
population ([Sittler and Strobel 1987]); such a small anisotropy is expected
to
have negligible consequences on our results.
From Liouville's theorem, the velocity
distribution is constant along particle trajectories, so that
the distribution at distance z is 
with, from
conservation of energy
We consider only latitudes such that 
, where the potential is
attractive and monotonic (with B nearly constant), so that the
isotropy of the velocity distribution is preserved and the
trajectories at z connect to z = 0 (the problem of accessibility
in phase space should be considered for higher latitudes,
where the potential is not monotonic). We thus have
with
The moment of order q of the velocity distribution at distance z
along 
is
The density 
is the moment of order q = 0, i.e., 
. In
general, 
is a decreasing function of the velocity; as a
consequence, since 
increases monotonically
with z,
decreases with z. Hence, all
the moments 
decrease with z; this is true in particular of the density.
In the absence of local thermal equilibrium, the concept of
``temperature" is not straightforward, and different types of
measurements can give different results. So we define
generalized temperatures 
as
This normalization has been chosen in such a way that all 's
are equal to 
if the distribution is a Maxwellian
of temperature , i.e.
On the other hand, for a
non-Maxwellian distribution, the temperatures are different.
In the non-Maxwellian case, the ``temperature" is
traditionally defined as the mean random energy times 
,
which is just our generalized temperature of order q = 2, i.e., 
. This
traditional definition is adequate when the physics and/or the
measuring device are sensitive to the mean random energy of the
particles.
However, many ``temperature" measuring techniques are sensitive
instead to different moments of the distribution, i.e., to
other . For example, a measurement of the Debye length
would give the effective temperature 
, since 
; on the
other hand, a measurement of the random flux would give
, since the mean random velocity is 
.
