4 Ionization equilibria

Figure 17: Mean electric charge versus temperature for oxygen (top panels) and iron (bottom panels) for different electron distributions. Black thick lines: Maxwellian electron distribution. Black thin lines: Hybrid distribution with $\alpha =1.5$ (left panels), $\alpha =2$ (middle panels) and $\alpha =3$ (right panels). Three values of $x_{\rm b}$ are considered: $x_{\rm b}=10$ (dotted line), $x_{\rm b}=5$ (dashed line) and the extreme value $x_{\rm b}=\alpha +1/2$ (full line).

4.1 Collisional ionization equilibrium (CIE)

The ionization equilibrium fractions, for coronal plasmas, can be computed from the rates described in the previous sections. In the low density regime (coronal plasmas) the steady state ionic fractions do not depend on the electron density and the population density ratio N_Z,z+1/N_Z,z of two adjacent ionization stages Z^+(z+1) and Z^+z of element Z can be expressed by:

$$% \frac{N_{Z,z+1}}{N_{Z,z}}=\frac{C^{Z,z}_{I}}{\alpha^{Z,z+1}_{R}}$$ (18)

where C^Z,z_Iand $\alpha^{Z,z+1}_{R}$ are the ionization and total recombination rates of ion Z^+z and Z^+(z+1) respectively. To assess the impact of the Hybrid rates on the ionization balance, we computed the variation with temperature of the mean electric charge of the plasma. This variation is compared with the variation obtained for a Maxwellian electron distribution in Fig. 17 for oxygen and iron and for different values of the parameters $x_{\rm b}$ and $\alpha$.

As expected, the plasma is always more ionized for a Hybrid electron distribution than for a Maxwellian distribution. The mean charge at a given temperature is increased, since the enhancement of the ionization rate is always much more important than a potential increase of the dielectronic rate (e.g. compare Figs. 5 and 14). The effect of the Hybrid distribution on the plasma ionization state is thus governed by the enhancement of the ionization rates. The enhancement of the plasma mean charge is more pronounced for smaller values of $x_{\rm b}$ and smaller values of $\alpha$ (Fig. 17), following the same behavior observed for the ionization rates (due to the increasing influence of the high energy tail). Similarly the effect is more important at low temperature, and a clear signature of the Hybrid distribution is the disappearance of the lowest ionization stages, that cannot survive even at very low temperature. For instance, for $\alpha =2$ and the extreme corresponding value of $x_{\rm b}=\alpha +1/2$, the mean charge is already +4 for oxygen and +6 for iron at T = 10⁴ K. At high temperature, the mean charge can typically be changed by a few units, the effect being more important in the temperature range where the mean charge changes rapidly with temperature in the Maxwellian case.

Figure 18: Mean electric charge versus temperature for the Hybrid electron distribution (black thin lines) compared to the Maxwellian distribution (black thick lines). The slope parameter $\alpha$ of the Hybrid distribution is fixed to 2, and the $x_{\rm b}$ values are as in Fig. 17. Each panel corresponds to an element and is labelled accordingly.

The same behavior is seen for all elements (Fig. 18). One notes that the effect of the Hybrid distribution generally decreases with Z. Again this is a consequence of the same behavior observed on the ionization rates (see Fig. 7).

A remarkable effect of the Hybrid distribution is that the mean charge is not always a monotonous function of temperature, in the low temperature regime. This is clearly apparent in Figs. 17 and 18 for $10^4~{\rm K} \le T \le 10^5$ K and $x_{\rm b}=10$. This phenomenon can only occur when the dielectronic rate dominates the total recombination rate and in the temperature range where this rate increases with temperature. In that case, the density ratio of two adjacent ions, N_Z,z+1/N_Z,z, can decrease with temperature provided that the ionization rate of Z^+z increases less rapidly with temperature than the recombination rate of the adjacent ion Z^+(z+1) (Eq. (18)). This usually does not occur in the Maxwellian case, but can occur in the Hybrid case, due to the flatter temperature dependence of the ionization rates for this type of distribution. For instance, for $3\times 10^{4}~{\rm K} \leq T \leq 7\times 10^{4}$ K, the ionization rate of O⁺² is increased by a factor of 2.5 for an Hybrid distribution with $x_{\rm b}=10$ (Fig. 8), whereas the total recombination rate of O⁺³ is increased by a slightly larger factor of 2.7 (see the corresponding grey line in Fig. 16, as seen above for $x_{\rm b}=10$ the total rate is basically unchanged compared to the Maxwellian case). The mean charge, which is around $\langle z \rangle = 2.5$, is thus dominated by the behavior of these ions and decreases in that temperature range.

4.2 Non-equilibrium ionization (NEI)

Figure 19: Mean electric charge versus temperature for oxygen (top panels) and iron (bottom panels) for different ionization timescales (thin lines) compared to equilibrium (bold lines) and for two extreme electron distributions, namely Maxwellian (solid lines) and Hybrid (dotted lines) for $\alpha =2$ and $x_{\rm b}=\alpha +1/2$.

Collisional Ionization Equilibrium (CIE) is not always achieved. For example, in adiabatic supernova remnants, the ionization timescale is longer than the dynamical timescale, so that the plasma is underionized compared to the equilibrium case. In non-equilibrium conditions, the ionization state of the gas depends on the thermodynamic history of the shocked gas (temperature, density) and time elapsed since it has been shocked.

The time evolution of the ionic fractions is given by:

$$% \frac{{\rm d}X_{Z,z}}{{\rm d}t} = n_{\rm e} [C_{\rm I}^{Z,z-1}X... ...1} + {\alpha^{Z,z+1}_{R}}X_{Z,z+1}- (C_{I}^{Z,z} + {\alpha^{Z,z}_{R}})X_{Z,z} ]$$ (19)

 

$${\rm with}\quad X_{Z,z} = \frac{N_{Z,z}}{\sum_{i} N_{Z,i}}\cdot$$

To estimate the effects of a Hybrid electron distribution on the ionization in non-equilibrium ionization conditions, we assume that the gas has been suddently heated to a given temperature, which stays constant during the evolution. The ionization timescale depends then on $\int{n_{\rm e}}~{\rm d}t$, where $n_{\rm e}$ is the number density of electrons and t the time elapsed since the gas has been heated. Within this assumption, the coupled system of rate equations can be resolved using an exponentiation method (e.g., Hughes & Helfand 1985).

For different ionization timescales (up to equilibrium), we computed the variation with temperature of the mean electric charge of oxygen and iron in two extreme cases of the electron distribution: Maxwellian and Hybrid with $x_{\rm b}=\alpha +1/2$ and $\alpha =2$. For small ionization timescales ( $n_{\rm e}~t \simeq 10^{8}-10^9$ s cm^-3), the effect of the Hybrid distribution on the mean electric charge is small, it increases with the ionization timescale and is maximum at equilibrium as is illustrated for oxygen and iron in Fig. 19. As in the equilibrium case, the effect from non-thermal electrons is always more important at low temperature and vanishes at high temperature. Note that the mean electric charge is slightly larger at high temperature for the thermal population than for the non-thermal one, as a consequence of the decrease of the ionization cross section at very high energy.