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### Description

During the 1970's a hot controversy emerged between in-situ measurements of the electron temperature in the ionosphere and ground measurements by incoherent backscatter radars. We suggest a possible explanation to this controversy. We define the ``local'' temperature of ionized species by the variance of the square root of the energy, and here by:

$$ \frac{3}{2} n_\infty kT(\mathbf{r}) = \!\! \int_{(-2q\phi(\mathbf{r})/m)^{1/2}}^{+\infty} \!\! \left( q \phi(\mathbf{r})+\frac{1}{2} mv^2 \right) \left( \frac{m}{2\pi kT} \right)^{3/2} e^{-\left[ q\phi(\mathbf{r})+\frac{1}{2} mv^2\right] /kT(\mathbf{r})} \,\mathrm{d}^3\mathbf{v} . $$
Here $n_\infty$ is the ambient density of the ionized gas, $k$ the Boltzmann constant, $T$ the temperature, $q$ the charge, $\phi$ the electric potential at a point $\mathbf{r}$ and $m$ is the mass.
For repelled species, this equation has an analytical solution, and we obtain:
$$\frac{3}{2}kT(\mathbf{r}) = e^{-q\phi(\mathbf{r})/kT(\mathbf{r})} \left[ q\phi(\mathbf{r})+\frac{3}{2}kT(\mathbf{r}) \right] ,$$
with the conditions:
$$\lim_{\mathbf{r}\to+\infty} \phi(\mathbf{r}) = 0;\ \ \lim_{\mathbf{r}\to+\infty} T(\mathbf{r}) = T_\infty;\ \ \lim_{\mathbf{r}\to+\infty} n(\mathbf{r}) = n_\infty .$$
If the electric potential vanishes, the local temperatures of ionized species go towards their ambient temperatures. The theoretical implication is that the distribution function $f(E,T(\mathbf{r}))$ is not a constant along a trajectory in the phase space, and that the Vlasov equation is violated.
Like Laframboise and Parker (1973), we shall consider an ionized gas without collisions and the ideal case of a potential well with a spherical symmetry, but there is no physical body. We define the number density as:
$$ n = \int f\,\mathrm{d}^3\mathbf{v} = n_\infty \left( \frac{m}{2\pi kT(r)} \right)^{\!\!3/2} \!\! \exp\left( \frac{-q\phi(r)}{kT(r)} \right) \times \!\!\!\! \int_{(-2q\phi(r)/m)^{1/2}}^{+\infty} \!\!\!\! e^{-mv^2/2kT(r)}\ 4\pi v^2 \,\mathrm{d} v . $$
For repelled species; we generalize Boltzmann's formula:
$$ n(r) = n_\infty \exp\left[ \frac{-q\phi(r)}{kT(\phi(r),r)} \right] . $$
We now calculate particles fluxes for attracted and repelled species. This will be a generalisation of Laframboise and Parker's (1973) formulas:
$$ J = \int fv_r\,\mathrm{d}^3\mathbf{v} = J_0 \left( 1 - \frac{q\phi(a)}{kT(\phi(a),a)} \right) , $$
where $J_0$ is the thermal flux and $a$ the radius of a spherical probe. For the repelled species, we have:
$$ J = J_0 \exp \left[ \frac{-q\phi(a)}{kT(\phi(a),a)} \right] . $$

In other words, the current due to repelled species is no longer an exponential! Our results also imply a modification of the PIC simulation methods. Because the temperature is no longer considered as a parameter but as a variable.

Reference:

Laframboise, J.G. and Parker L.W. (1973), {\it The Physics of Fluids}, p. 629.