2 Collisional ionization and radiative recombination rate

2.1 Collisional and auto-ionization rate

To describe the ionization processes we refer to the work of Arnaud & Rothenflug (1985) and to the updating for the Fe ions of Arnaud & Raymond (1992) (hereafter AR85 and AR respectively). The contributions to the ionization rates are given for all the ions of H, He, C, N, O, Ne, Na, Mg, Al, Si, S, Ar, Ca and Ni from different atomic subshells separately. The direct ionization rate coefficients versus temperature T are given by:  

$$C_{\rm DI}={6.69 \ 10^{7}\over (kT)^{3/2}} {\sum_{j} {\exp (-x_{j})\over x_{j}}} F(x_{j}) \ \ \ [\mbox{cm}^3/\mbox{s}]$$ (1)

 

F(x_j)= A_j[1-x_jf₁(x_j)]+ (2)

+bj[1+x_j-x_j(2+x_j)f₁(x_j)] C_jf₁(x_j)+D_jx_jf₂(x_j).

The summation is performed over the subshells j of the ionizing ion with  

$$x_{j} ={I_j\over kT}, \ \ \ f_1(x)={\rm e}^x \int_{1}^{\infty} {{\rm d}t \over t} {\rm e}^{-tx},$$ (3)

$$f_2(x) ={\rm e}^x \int_{1}^{\infty} {{\rm d}t \over t} {\rm e}^{-tx}\ln (t),$$

and where kT and I_j are in eV. The values parameters A_j, B_j, C_j and D_j are given in AR85 and AR. Following AR85 and AR we also take into to account the excitation-autoionization (hereafter EA) contribution of ions in the ground state. This is a good approximation for low-density plasmas (as in supernova remnants or clusters of galaxies) because the lifetime of excited states is small as compared to the mean collision time. The other ions not included in the AR85 work are calculated by interpolation or extrapolation along the isosequence.

2.2 Radiative recombination rate

For the radiative recombination rates¹ the calculations of Shull & Van Steenberg (1982) (hereafter SV) were used for some of the most abundant astrophysical elements (Mg, Si, S, Ar, Ca, Fe, Ni). They give the fitting parameters A and $\eta$ for the following formula:  

$$\alpha_{\rm r}=A{\left( {T\over 10^4 {\rm K}} \right)^{\eta}} \ \ \ [\mbox{cm}^3/\mbox{s}],$$ (4)

with the electron temperature T in eV. LM extrapolated these calculations also to other astrophysical less abundant elements. For Fe XV-Fe XXIV we used the formula of AR:  

$$\alpha_{\rm r}=A{\left( {T\over 10^4\rm K} \right)^{\alpha-\... ...t( {T\over 10^4\rm K} \right) }} \ \ \ [\mbox{cm}^3/\mbox{s}],$$ (5)

where the electron temperature T is in Kelvin and the fitting parameters $A, \alpha, \beta$ are given by AR in tabular form. For the H-like, He-like, Li-like and Na-like isosequences we used the new calculations, in the framework of the opacity project, of Verner & Ferland (1996). They fit the data by the following formula:  

$$\alpha_{\rm r}=A\left[{\sqrt{T\over T_0}\left(1+ \sqrt {T\ov... ...ht)^{1-b} \left(1+ \sqrt{T\over T_1} \right)^{1+b}}\right]^{-1}$$ (6)

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ... ... \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [\mbox{cm}^3/\mbox{s}],$$

with the electron temperature T in Kelvin, and the fitting parameters A,b, T₀, T₁ are given for all the ions from H through Zn. This last formula is valid in a wide range of temperatures, from $T=2.5 \ 10^{-4}$ eV to T=100 keV. We also use the same formula for C, N, O and Ne ions with the fitting parameters of Péquignot et al. (1991).