3.2 Basics of the Model

In discussing the notion of the centrifugal equator, we mentioned the component of the centrifugal force (due to the corotation of the plasma) that is parallel to the magnetic field. This force is proportional to the mass of the particle species considered and is consequently determined by the motion of the ions. From a microscopic view, in order for the plasma to remain neutral, the electrons must suffer an appropriate electromotive force to ``follow'' the motion of the ions. This force is produced by an ambipolar electric field $\vec{E}$ parallel to the magnetic field. This ambipolar electric field changes sign at the centrifugal equator and is derived from a negative electrostatic potential (denoted by $\phi _E$) in order to confine negative charges. To first approximation, the corresponding potential energy $\Phi_e=-e \phi_E$ confines the electrons about the centrifugal equator to the same extent that the ions are confined by the combination of the centrifugal potential and the electric potential energy $Z_{i}e\phi_E$ (where $Z_{i}$ designates the charge state of the ion).

One can see that the potential $\Phi_e$, needed to calculate the electron profile, depends fundamentally on the ions (via the equation for plasma neutrality), and on their velocity distributions as well as their chemical composition (see equations A1,A2 in appendix). Since we only have electron measurements from Ulysses we need to look elsewhere for information about the ions. Unfortunately, the separate velocity distributions of the ion species in the torus ( ${\rm S^+, S^{++}, S^{+++}}$, ${\rm O^+, O^{++}}$, ${\rm H^+}$, ${\rm Na^+}$ and ${\rm SO_2^+}$) are not well determined by the Voyager PLS instrument in the outer, warm region of the IPT. However, there is strong evidence that both the electron and ion distributions are non-Maxwellian [ Bagenal and Sullivan, 1981, Sittler and Strobel, 1987]. Furthermore, the mean-free-path of the ions (which are about 10 times hotter than the electrons) is also much larger than that of the electrons which argues for a still less effective thermalization for the ions [ Smith and Strobel, 1985]. Thus, there is every reason to suppose that the ions are not in local thermal equilibrium, just as the Ulysses data showed that the electrons are not thermalized.

In their analytical model describing the velocity distribution of the ions with a kappa function, Meyer-Vernet, Moncuquet and Hoang [1995] made the simplifying assumptions of a single ion species having a similar (isotropic) distribution as the electrons, which is certainly not justified, as they frankly admit. This limitation can be over come by calculating separate density profiles using a set of equations (4), with a potential energy $\Phi_e=-e \phi_E$ for the electrons and $\Phi_i = Z_ie\phi_E + \phi_C$ for each ion species of charge $Z_ie$ (where $\phi_C$ is the centrifugal potential -see equation (A2) in appendix). This involves a set of 9 equations (1 for electrons and 8 for the ion species listed above) with 10 unknowns (the densities and the ambipolar electrostatic potential $\phi _E$), which is closed by the equation of charge neutrality $\sum_i {n_i Z_i} = n_e$.