The current-voltage relationship revisited: exact and approximate formulas with almost general validity for hot magnetospheric electrons for bi-Maxwellian and kappa distributions

We derive the current-voltage relationship in the auroral region taking into account magnetospheric electrons for the bi-Maxwellian and kappa source plasma distribution functions. The current-voltage formulas have in principle been well known for a long time, but the kappa energy ̄ux formulas have not appeared in the literature before. We give a uni®ed treatment of the bi-Maxwellian and kappa distributions, correcting some errors in previous work. We give both exact results and two kinds of approximate formulas for the current density and the energy ̄ux. The ®rst approximation is almost generally valid and is practical to compute. The ®rst approximation formulas are therefore suitable for use in simulations. In the second approximation we assume in addition that the thermal energy is small compared to the potential drop. This yields even simpler linear formulas which are suitable for many types of event studies and which have a more transparent physical interpretation than the ®rst approximation formulas. We also show how it is possible to derive the ®rst approximation formulas even for those distributions for which the exact results can not be computed analytically. The kappa ®eld-aligned conductance value turns out always to be smaller than the corresponding Maxwellian conductance. We also verify that the obtained kappa current density and energy ̄ux formulas go to Maxwellian results when j!1.