5 Conclusions

We have studied the effect on the ionization and recombination rates, as well as on the ionization balance, of a non-thermal electron distribution, as expected in the vicinity of strong shocks.

The electron distribution is modelled by a Maxwellian distribution at low energy up to a break energy, and by a power-law distribution at higher energy. It is caracterised by the three parameters kT(the temperature of the Maxwellian part), $x_{\rm b}$ the reduced energy break, and $\alpha$ the slope of the power-law component. We only considered the parameter range where $x_{\rm b}>\alpha+1/2$ which corresponds to an enhanced high energy tail. All the behaviors outlined are only valid for this range of parameters.

We provide exact formulae of the ionization rates for this Hybrid electron distribution in the Appendix, and approximate estimates of the radiative recombination rates (Eqs. (12) and (13)) and of the dielectronic recombination rates (Eq. (16)). The Hybrid rates depend on the ion considered and on the parameters kT, $\alpha$ and $x_{\rm b}$. Computer codes are available on request.

For the parameter range considered, the proportion of electrons at high energies and the mean energy of the distribution is a monotonic function of $x_{\rm b}$ and $\alpha$. As expected, the modification of the rates for the Hybrid distribution, as compared to the Maxwellian distribution of the same temperature, increases with decreasing $x_{\rm b}$(with a threshold at about $x_{\rm b}\sim 10{-}20$, higher for lower temperature) and decreasing $\alpha$.

The impact of the Hybrid electron distribution on the ionization rates depends on how the high energy tail affects the proportion of electrons above the ionization potential $E_{\rm _{\rm I}}$. The Hybrid rates are increased with respect to the Maxwellian rates except at very high temperature. The enhancement factor depends on the temperature, mostly via the factor $E_{\rm _{\rm I}}/kT$, and increases dramatically with decreasing temperature. For a given ion, it is always important at T^*, the temperature of maximum ionization fraction for a Maxwellian distribution under ionization equilibrium, where it can reach several orders of magnitude.

The effect of the hybrid distribution on the dielectronic rate depends on the position of the resonance energies $E_{\rm _{\rm DR}}$ as compared to the power-law energy break. The dielectronic rate can only be increased if $kT < E_{\rm _{\rm DR}}/(\alpha+1/2)$. At T^* the enhancement factor is typically less than an order of magnitude. At high temperature, the dielectronic recombination rate is slightly decreased (by typically $10\%$ at most). The effect of the hybrid distribution on the radiative recombination rates is only of the order of a few $10\%$ at most.

The ionization balance is affected significantly, whereas the effect is smaller in ionizing NIE plasmas. The plasma is always more ionized for a Hybrid electron distribution than for a Maxwellian distribution. The effect is more important at low temperature, and a clear signature of the Hybrid distribution is the disappearance of the lowest ionization stages, which cannot survive even at very low temperature.

Acknowledgements

We would like to thank Jean Ballet for a careful reading of the manuscript.