A. Appendix: The Geometrical and Numerical Method

The geometry of a tilted dipole field is shown in Figure A1. The centrifugal equator is defined as the locus of the most distant points from the spin axis along magnetic field lines. The centrifugal equator is plotted here in the Jovian meridian plane which contains the magnetic moment $\vec{\cal{M}}$ of a centered dipole, tilted by $\alpha$ with respect to the Jupiter spin axis $\vec{\Omega}$. The angle $\psi$ between the magnetic and centrifugal equator is maximum in this $(\vec{\Omega},\vec{\cal{M}})$ plane and is about $\alpha/3$ for small $\alpha$[ Hill et al., 1974] (for Jupiter $\psi \sim 3.2$$^\circ$). A given point P of the magnetosphere is defined by two coordinates: (i) the dipole equatorial radius L of its own magnetic field line and (ii) its centrifugal latitude, that is the angle between OP and the centrifugal equator.

Figure A1: Geometry of a tilted dipole field.

In construction of a 2-D model of the spatial distribution of a plasma with N particle species we need to determine the properties at each point $P$ (described by its value of the curvilinear co-ordinate along the magnetic field) using a system of N+1 equations (the equations of density (10) plus charge neutrality) with N+1 unknowns (the densities and the ambipolar electric potential $\phi _E$ at point $P$).

$$\sum_{\alpha}^{\rm N} n_{\alpha}\left(s,\Phi_\alpha(s,\phi_E)\right) Z_{\alpha} = 0$$ (13)

Where $\Phi_\alpha$ is the energy potential at point $P$, the sum of centrifugal, gravitational and electrostatic potential energies:

$$\Phi_\alpha(s,\phi_E) = \frac{m_\alpha \Omega_J^2}{2} ({\rm x^2_{max}} - {\rm x}^2 )$$

$$\, + m_\alpha G M_J (\frac{1}{r}-\frac{1}{r_0}) + Z_\alpha e \phi_E$$ (14)

where $\Omega_J$ is the rotation frequency of Jupiter, $x$ the cylindrical distance of $P$ from the rotation axis of Jupiter, $G$ is the gravitational constant and $M$ is the mass of Jupiter (N.B. the gravitational potential is very small compared with the other terms). For the field line threading point $P$, ${\rm x_{max}}$ and $r_0$ are the distances of the centrifugal equator from the rotation axis and center of Jupiter respectively.

Solving the set of equations using Newton's method, we pose

$$F^{(0)}=\sum_{\alpha}^{\rm N} n_{\alpha}(\phi^{(0)}_{E}) Z_{\alpha}$$ (15)

and obtain the electric potential $\phi _E$ by iteration of :

$$\phi^{(n+1)}_{E} = \phi_E^{(n)} + F^{(n)}/(\partial F^{(n)}/ \partial \phi^{(n)}_E)$$ (16)

For bi-Maxwellian distributions the convergence is very rapid ( $\partial F/ \partial \phi_E$ is linear). The bi-kappa distributions require about 25 iterations to converge. The electric ambipolar potential $\phi _E$ is typically on the order of $\approx -50 V$ at 9 ${\rm R_J}$ and 10$^\circ$centrifugal latitude. We show in figure A2 the isocontours of $\phi _E$ as found for the density/temperature models of figures 7, 8 and 9(left panels).

Figure A2: Contours sample, in a meridian plane, of the ambipolar electric potential $\phi _E$ (labelled in volt).

Finally, note that the kappa thermal speeds $\Theta_\parallel$ (needed for solving equation 10) are derived from the parallel and perpendicular temperatures of the core and halo components of the ions $n_c, T_c, n_h, T_h$, using

$$\frac{m \Theta^2_\perp}{2k_B}\frac{\kappa}{\kappa - 3/2}= \frac{n_c T_c + n_h T_h}{n_c + n_h}$$ (17)

and that $A_0=\Theta_{\perp}^2/\Theta_{\parallel}^2$.