Space Charge

Space charge trapped in a magnetic field is equivalent to field-aligned potentials, and can exist in a neutral plasma wherever the pitchangle distributions of ions and electrons are unequal [#!Alfven63!#,#!Persoon65!#,#!Whipple77!#]. That is, one can envision the space charge electric field, set up by unequal pitchangle distributions, as reacting on the ion and electron pitchangles so as to force them to co-exist in equal densities. Thus the expected electric field is a function of the anisotropy and thermal energy of the plasmas involved, where the maximum voltage expected is a small multiple of the thermal energy. Now presumeably the electron stripping or ion collisional source of our trapped ions produces an ion thermal energy of a few volts, far too small to account for the observed $\sim 100$ V potentials observed. The electrons that stream along the magnetic field, on the other hand, can be accelerated by the vacuum electric field to some fraction of 400 V. However, one would expect that the same electrons would rapidly short out the vacuum field, leading to a chicken-and-egg problem concerning the origin of the steady state $\sim 200$ V parallel potential.

That is, since diffusion and collisions tend to thermalize and isotropize all the charged particles, one would expect the space charge to dissipate rapidly. But experimentally both the potential and the space charge appear to be quasi-permanent suggesting we have a non-equilibrium or steady-state plasma solution with sources and sinks. In [#!Sheldon01!#] we argue that hot trapped ions when neutralized by field-aligned electrons lead to an anisotropic field-aligned potential proportional to the temperature of the ions. As long as the anisotropy is continuously maintained against the losses due to scattering, the voltage can be sustained indefinately. The difficulty here is in identifying this continuous source of the hot ions.

Based on our earlier discussion about the annular plasma gap, we argue that ions produced by electron stripping, $S_e$, on high magnetic latitude field-lines will diffuse toward the magnet simultaneously gaining energy and becoming more equatorially trapped. If the first adiabatic invariant, $\mu = E_\perp/B$, is relatively conserved, then the energy of the ions is proportional to the magnetic field strength during this transport. Since $B= B_0/r^3$, a radial transport of only a 6 magnet radii, from 5 cm to 2 cm will produce an energy gain of 216, easily accounting for the $\sim 200$ V potentials observed.

The model then, is as follows. Ions are produced at large radial distances from the magnet by electron impact ionization generated by electrons streaming from the magnet at high magnetic latitudes. These ions diffuse inwards by ion-neutral collisions and gain energy by conservation of the first adiabatic invariant. This produces a space charge which is neutralized by electrons emitted from the magnet. However, the high anisotropy differences between the near-90$^\circ$ trapped ions and the field-aligned electrons generate a field-aligned potential whose magnitude approaches the perpendicular energy of the hot ions, which are also responsible for the hot parallel temperature of the electrons. Continual energy losses of ions are offset by a steady input from large distances resulting in a steady state field-aligned potential drop. This steady state system can be observed at high neutral gas pressures (see figure 5) as a pink annulus of glowing ions sandwiched between two domes of glowing electrons.