2.1 Velocity Filtration

In order to explain the increase in electron temperature with latitude and anti-correlation with density observed by Ulysses along its north-south traversal of the torus, Meyer-Vernet, Moncuquet and Hoang [1995] invoked non-Maxwellian velocity distributions and the ``velocity filtration'' mechanism (as first proposed by Scudder [1992a,b] in a different context). The underlying principle is that the plasma is subjected to an attractive potential (in this case the centrifugal force due to corotation) and is not in thermodynamic equilibrium. The low energy particles are confined in the potential well (which defines the centrifugal equator) whereas the more energetic particles can escape more easily. The farther out of the well, the larger the proportion of energetic particles and, hence, the higher the average kinetic energy. The temperature therefore increases with centrifugal latitude as the density decreases. This should be a general property of dilute planetary plasmaspheres and plasma tori (see [ Meyer-Vernet, 2001] for a discussion at a basic level).

We shall thus consider here charged particles confined by a corotating magnetic field $\vec{B}$ and examine the distribution as a function of position (curvilinear co-ordinate $s$ along the magnetic field line with the origin located at the centrifugal equator -see the figure in appendix). Let $f_0(v)$ be the velocity distribution (which we assume in this section to be isotropic for simplicity) at $s=0$ and suppose that the particles experience the force associated with an attractive potential ($\Phi(s) > 0$ with a minimum at $s=0$). Liouville's theorem states that the velocity distribution is constant along a particle trajectory and, as a consequence, the velocity distribution at $s$ is $f(s,v)=f_0(v_0)$. Conservation of energy requires that $v^2 = v_0^2 -2\Phi/m$. One then finds that

$$f(s,v) = f_0\left(\sqrt{v^2 + 2\Phi(s)/m}\right)$$ (1)

(assuming that $s$ is accessible in phase space, as it is in this case of a monotonic and attractive potential). The moment of order $q$ of the distribution at $s$ along the field line is

$$M_q(s) = \int{v^q f(s,v) d^3v}$$

$$= 4\pi \int_0^\infty {v^{2+q} f_0\left(\sqrt{v^2 + 2\Phi(s)/m}\right) dv}$$ (2)

In general, $f_0$ is a decreasing function of velocity and, since the potential $\Phi(s)$ is monotonically increasing, the distribution $f_0(\sqrt{v^2 + 2\Phi(s)/m})$ decreases with $s$. Consequently, all the moments decrease with $s$, in particular the density $n=M_0$.

If $f_0$ is a single Maxwellian of temperature $T$, one can see from (1) that the distribution remains a Maxwellian for $s \neq 0$, and from (2) that all the moments behave similarly with $s$. The temperature $T = m M_2/3k_B M_0$ or the effective temperature $T_{eff} = m M_0/k_B M_{-2}$ (which was obtained by a spectroscopic analysis of Bernstein modes measured by Ulysses [ Moncuquet et al., 1995]) thus remain constant. This means that with a Maxwellian distribution at the centrifugal equator the potential filters all the particles in the same way. In contrast, if the distribution $f_0$ has a suprathermal tail, i.e. more energetic particles than a Maxwellian should have, the higher moments decrease less rapidly with $s$ than does the zero-order one, so that the temperature increases with $s$. This generic temperature inversion was first shown in a graphic way by Scudder [1992a] and proved analytically for any linear combination of Maxwellians [ Meyer-Vernet, Moncuquet and Hoang, 1995].