2.3 Maxwellian Core and Halo Velocity Distributions

The Voyager 1 electron measurements have been interpreted by Sittler and Strobel [1987] in terms of a superposition of two Maxwellian distributions with different temperatures and densities (`` core + halo ''). Such a distribution does lead to a temperature increase along the magnetic field as the density decreases.

Let us take a function that is a sum of two Maxwellian distributions of densities $n_c,n_h$ and temperatures $T_c, T_h$, corresponding to the core (cold) and halo (hot). With such a distribution, one can define the traditional temperature as $T=(n_cT_c+n_hT_h)/(n_c+n_h)$ and the effective temperature as $T_{eff}=(n_c+n_h)/(n_c/T_c+n_h/T_h)$. For simplicity, we will write $\alpha = n_h/n_c$ and $\tau = T_h/T_c$ (which implies $0\le \alpha < 1$ and $\tau > 1$). In this case velocity filtration is especially simple to visualize since the density of the cold electrons $n_c$ at the equator is multiplied by a factor $\beta(s)=e^{-\Phi(s)/k_BT_c}$ at a distance $s$ and similarly for hot electrons. One obtains the variation in temperatures with latitude for such a core + halo distribution to be:

$$T(s) = \frac{1+\beta(s)^{\frac{1}{\tau}-1}\alpha\tau}{1+\beta(s)^ {\frac{1}{\tau}-1}\alpha} T_c$$ (6)

$$T_{eff}(s) = \frac{1+\beta(s)^{\frac{1}{\tau}-1}\alpha} {1+\beta(s)^{\frac{1}{\tau}-1}\alpha/\tau}T_c$$ (7)

For Voyager 1, at the distance of $\sim 8 {\rm R_J}$, we have $\alpha\approx 0.02$ and $\tau\approx 12$, and if the conditions were the same for both Voyager 1 and Ulysses, the above equations allow us to predict that a factor of 4 variation in density should have produced a 50% variation in $T$ and a 5% variation in $T_{eff}$. In reality, Ulysses measured a doubling of effective temperature for a factor 4 drop in $n_c$. The variation in $T_{eff}$ is very small for a core + halo distribution because $T_{eff}$ is close to $T_c$ which is constant along a field line since the core is Maxwellian.