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
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 .