2.2 Polytropic Law and Kappa Velocity Distributions

Let us first consider the simple case treated by Meyer-Vernet, Moncuquet and Hoang [1995], which we will generalize in this paper, of the following (normalized) $\kappa$-distribution

$$f_{0}\left( v \right) = \frac{\Gamma(\kappa +1)}{\pi^{3/2}\kappa^... ...Theta^3} \left[ 1+\frac{v^{2}}{\kappa \Theta^{2}} \right]^{-\kappa -1} \protect$$ (3)

where $\Theta$ is the most probable speed, which we will denote the ``thermal-kappa speed'', in an analogy to the thermal speed of a Maxwellian. Kappa distributions have been widely used to model space plasmas [ Vasyliunas, 1968, Collier and Hamilton, 1995] and approximate a Maxwellian at low energies with a power-law tail at higher energies (a non-thermal halo). Note that typically kappa distributions are found to have a $\kappa$ index between 2 and 6 and that the distribution tends towards a Maxwellian when $\kappa \mapsto \infty$. In this limit the two temperatures, $T$ and $T_{eff}$, are equal but for finite $\kappa$ they have the different values $T = (m \Theta^2)/(2k_B)\times \kappa/(\kappa - 1.5)$ and $T_{eff}= (m \Theta^2)/(2k_B)\times \kappa/(\kappa - 0.5)$, though they both increase with latitude and obey the same polytropic law. To calculate the variation of these temperatures with position $s$ along the magnetic field line, one inserts (3) in (1) and finds that the distribution remains a kappa function (with the same $\kappa$). Using moments calculation, one finds the density and temperature profiles to be:

$$\frac{n(s)}{n(0)} = {\left[ 1+\frac{2\Phi(s)}{m \kappa \Theta^2} \right]} ^{\frac{1}{2} - \kappa} \protect$$ (4)

$$\frac{T(s)}{T(0)} = \frac{T_{eff}(s)}{T_{eff}(0)} = \left[\frac{n(s)}{n(0)}\right] ^{\frac{-1}{\kappa - 1/2}}$$ (5)

Thus, with such a $\kappa$-distribution, the density and temperature follow a polytropic equation of state $T \propto n^{\gamma - 1}$, where the polytropic index is less than one and related to $\kappa$ via $\gamma = 1 - 1/(\kappa - 0.5)$ [ Meyer-Vernet, Moncuquet and Hoang, 1995].

The polytropic law ($\gamma = 0.48$) observed in the IPT suggests that we should model the electrons in the torus with a $\kappa$-distribution of $\kappa_e \approx 2.4$. The measurements do not tell us however that the ``true'' electron velocity distribution is necessarily a kappa. Moreover, the value that we derive for the polytropic law comes from measurements made along the specific magnetic field line sampled by Ulysses (about $L=8{\rm R_J}$). The distribution on neighboring field lines does not necessarily have the same value of kappa or even maintain a kappa-like distribution.