4 Atomic data and improvements

The intensities of the three lines (resonance, forbidden and intercombination) are calculated mainly from atomic data presented in Paper I (Porquet & Dubau 2000). In this work (as well as in Paper I) for all temperatures (low and high), radiative recombination contributions (direct + upper-level radiative cascades), and collisional excitations inside the n=2 shell were included in the line ratio calculations. For high temperatures, the collisional excitation contribution (direct + near-threshold resonance + cascades) from the ground level (n=1 shell, 1s²) are important as well as dielectronic recombination (direct + cascades).

Excitation collisional data are also taken from Paper I, which are based on the calculations from Zhang & Sampson (1987) plus the contribution of the upper-level (n>2) radiative cascades calculated in Paper I (see Paper I for more details).

The ionization balance is from Mazzotta et al. (1998) and the data for radiative and dielectronic recombinations are from Paper I. Various new data for the transition probabilities (e.g., forbidden and intercombination lines) have been selected (see Sect. 4.1).

In the following paragraphs we describe the several differences between Paper I and this work: A_ki, optical depth, contribution of the blended dielectronic satellite lines, and radiation field.

  
4.1 Update of the A $\mathsfsl{_{ki}}$ for the forbidden and the intercombination lines

 

 

Table 2: Update of the transition probabilities (s^-1) with published experimental values for the forbidden line (z, $A_{1\to 2}$) and the intercombination line (y, $A_{1\to 4}$) compared to the theoretical values from Porquet & Dubau (2000).

ion	forbidden line	intercombination line
	(z, $A_{1\to 2}$)	(y, $A_{1\to 4}$)
C V	4.857(+1) (S94)	2.90(+7) (H85)
N VI	2.538(+2) (N00)	1.38(+8) (H85)
O VII	1.046(+3) (C98)	5.800(+8) (E81)
Ne IX	1.09(+4) (T99)	5.400(+9) *
Mg XI	7.347(+4) (S95)	3.448(+10) (A81)
Si XIII	3.610(+5) *	1.497(+11) (A79)

(A79): Armour et al. (1979)
(A81): Armour et al. (1981)
(C98): Crespo López-Urrutia et al. (1998)
(E81): Engström et al. (1981)
(H85): Hutton et al. (1985)
(N00): Neill et al. (2000)
(S94): Schmidt et al. (1994)
(S95): Stefanelli et al. (1995)
(T99): Träbert et al. (1999)
^*: Theoretical values taken from Lin et al. (1977), see text (Sect. 4.1).

We have updated the transition probabilities A_ki reported in Paper I for the intercombination (y, $A_{1\to 4}$) and the forbidden (z, $A_{1\to 2}$) line by published experimental values (see Table 2, and references therein). In some cases, no published experimental values have been found and then we used the theoretical values from Lin et al. (1977). Indeed, comparisons of their theoretical values with the experimental values reported in Table 2 seem to show good agreement in other cases.

For C V, the ratio R is reduced by about 20$\%$ comparing the calculations using the values of A_ki reported in Paper I, while for N VI the reduction is less than $10\%$. For O VII, Ne IX, Mg XI, and Si XIII, the differences between the current calculations using these new values of A_ki and those reported in Paper I are negligible.

4.2 Influence of the optical depth (resonant scattering)

Schrijver et al. (1995) and Mewe et al. (1995b) have investigated the possibility that resonance photons are scattered out of the line of sight in late-type stellar coronae (see also Acton 1978). Indeed, in this process, a resonance line photon is absorbed by an ion in the ground state and then re-emitted, generally in a different direction. So, the total photon intensity integrated over 4$\pi$ remains unchanged but the photon distribution with respect to a given direction is altered. This absorption and re-emission is indistinguishable from scattering and depends on the geometry of the region being observed. In general, photons would be scattered preferentially out of the line of sight for active regions (relatively dense areas) and into the line of sight for the surrounding quiet Sun (less dense area), see Schmelz et al. (1997) and Mewe et al. (2001). The effect is smaller for instruments with a larger field of view.

This could have an impact on the temperature diagnostic, the so-called G=(z+x+y)/w or (f+i)/r ratio. If the optical depth of the line is not taken into account, the calculated intensity ratio G can be overestimated and the inferred temperature from the G ratio is underestimated.

As detailed in Mewe et al. (2001), branching ratios can be used to check the assumption of the optical thin model because effects of resonance scattering would affect the measured branching ratio. From the fact that the intensities of e.g., the strong resonance lines Fe XVIII$\lambda$93.92 and Fe XIX$\lambda$108.307 are in good agreement with the intensities of other lines sharing the same upper level, one can derive a constraint on the optical depth taking into account the systematic uncertainties of the theoretical transition probabilities A (typical 25$\%$ for each A, hence 35$\%$ for the branching ratio) which dominate over the statistical errors (typically 10$\%$). If we rule out a reduction in the resonance line intensity larger than about 30$\%$, then on the basis of a simple "escape-factor'' model with

$$% P(\tau) \simeq \frac{1}{[1+0.43 \tau]},$$ (22)

the escape factor for a homogeneous mixture of emitters and absorbers in a slab geometry (e.g., Kaastra $\&$ Mewe 1995), one can put a constraint on the optical depth. The optical depth $\tau$ for a Doppler-broadened resonance line can be written as (Mewe et al. 1995b):

 

$$% \tau 1.16\times 10^{-17}~\left(\frac{n_{i}}{n_{el}}\right)A_{Z}\left(\frac{N_{\rm {H}}}{n_{\rm e}}\right)\lambda f \sqrt{\frac{M}{T_{\rm e}}} n_{\rm e}l$$

$$\equiv 10^{-19}~C_{d}~\left(\frac{A_{Z}}{A_{Z_{\odot}}}\right)~\left(\frac{n_{\rm e}l}{\sqrt{T_{6}}}\right),$$ (23)

where ( n_i/n_el) is the ion fraction (e.g. from Arnaud & Rothenflug 1985, from Arnaud & Raymond 1992 for iron, or from Mazzotta et al. 1998), A_Z is the elemental abundance relative to hydrogen, $A_{Z_{\odot}}$ the corresponding value for the solar photosphere as given by Anders & Grevesse (1989), $N_{\rm H}/n_{\rm e}\simeq0.85$ the ratio of hydrogen to electron density, $\lambda$ is the wavelength in Å, f the absorption oscillator strength, M is the atomic weight, $T_{\rm e}$ is the electron temperature (in K or T₆ in MK), l a characteristic plasma dimension (in cm) and

$$% C_{\rm d}\equiv 98.5 \left(\frac{n_{i}}{n_{el}}\right) A_{Z_{\odot}} \lambda f \sqrt{M} .$$ (24)

According to Eq. (23), Ness et al. (2001a) estimated the optical depth, adopting a value of unity for the fractional ionization and using solar abundances. One further assumes $T_{\rm e}$ at the peak line formation, but note that $\tau$ is rather insensitive to the precise value of $T_{\rm e}$. One can determine - for each resonance line - that value of $n_{\rm e} \ell$ which yields an optical depth of unity. According to the values of n$_{\rm e}$ inferred from the ratio R=z/(x+y) or R=f/i (from C V to Si XIII the intercombination and the forbidden lines are not sensitive to resonant scattering below a column density of ${\cal N}_{\rm H}\sim 10^{25-26}$ cm^-2 and ${\cal N}_{\rm H}\sim 10^{30-31}$ cm^-2, respectively, while the resonance line becomes sensitive to the resonant scattering above ${\cal N}_{\rm H}\sim 10^{21-23}$ cm^-2), one can determine the corresponding values of $\ell$. One can compute the respective emission measures of $n_{\rm e}^2\ell^{\,3}$ respectively, and can compare these emission measures with those derived from the measured line fluxes $f_\lambda$ according to

$$% EM=\frac{4\pi d^{\, 2} f_{_{\lambda}}}{P_{_{\lambda}}(T_{\rm e})}$$ (25)

with the line emissivity $P_\lambda(T_{\rm e})$ and the distance d of the star. If the former is larger, this inconsistency shows that the assumption of a non-negligible optical depth is invalid and we conclude that optical depth effects are irrelevant for the analysis of He-like triplets. On the contrary, the effect of resonant scattering should be taken into account when comparing the theoretical values with the observational ones.

Since $\tau_{_{\rm r}}>>\tau_{_{\rm i}}$ ( $\tau_{_{\rm r}}$ and $\tau_{_{\rm i}}$ corresponding respectively to the optical depth of the resonance and the intercombination lines), we can write $G_{\tau}\equiv \frac{G}{P_{_{\rm r}}}$, where $G_{\tau}$ is the value of the ratio taken into account the optical depth of the resonance line, G is the value without resonant scattering (such as in Paper I and Sect. 5), and $P_{_{\rm r}}$ is the escape probability for the resonance line (Eq. (22)). One should note that $G_{\tau}$ is not strictly exact when the contribution of the blended dielectronic satellite lines are introduced in the calculations (see Sect. 4.3).

  
4.3 Blended dielectronic satellite lines

The intensity of a dielectronic satellite line arising from a doubly excited state with principal quantum number n in a Lithium-like ion produced by dielectronic recombination of a He-like ion is given by:

$$% I_{\rm s}=N_{\rm He}~ n_{\rm e}~ C_{\rm s},$$ (26)

where $N_{\rm He}$ is the population density of the considered He-like ion in the ground state 1s² with statistical weight g₁ (for He-like ions g₁=1).

The rate coefficient (in cm³s^-1) for dielectronic recombination is given by (Bely-Dubau et al. 1979):

$$% C_{\rm s}=2.0706\times 10^{-16}~\frac{{\rm e}^{-E_{\rm s}/kT_{\rm e}}}{g_{1} T_{\rm e}^{3/2}}~F_{2}(s),$$ (27)

where $E_{\rm s}$ is the energy of the satellite level s with statistical weight $g_{\rm s}$ above the ground state of the He-like ion, $T_{\rm e}$ is the electron temperature in K, and F₂(s) is the so-called line strength factor (often of the order of about 10¹³ s^-1 for the stronger lines) given by

$$F_{2}(s) = {{g_{\rm s} A_{\rm a} A_{\rm r}} \over {(A_{\rm a} + \sum A_{\rm r})}},$$ (28)

where $A_{\rm a}$ and $A_{\rm r}$ are transition probabilities (s^-1) by autoionization and radiation, and the summation is over all possible radiative transitions from the satellite level s.

For a group of satellites with the same principal quantum number n, $E_{\rm s}$ can be approximated by

$$% E_{\rm s} [\rm eV] = 1.239842\times 10^4 \ {a_{DR}\over {\lambda}},$$ (29)

where $\lambda$ is the wavelength (Å) of the satellite line and $a_{\rm DR} \simeq$ 0.7, 0.86, 0.92, and 0.96 for n = 2, 3, 4, and > 4, respectively (Mewe & Gronenschild 1981). For $\lambda$ in Å and T in K we can write:

$$% {E_{\rm s}\over kT} = {a_{\rm DR} hc\over {\lambda kT}} = 1.439\times 10^8 \ {a_{\rm DR}\over {\lambda T}}\cdot$$ (30)

The influence of the blending of dielectronic satellite lines for the resonance, the intercombination and the forbidden lines has been taken into account where their contribution is not negligible in the calculation of R and G, affecting the inferred electron temperature and density. This is the case for the high-Z ions, i.e. Ne IX, Mg XI, and Si XIII (Z=10, 12, and 14, respectively). Since the contribution of the blended dielectronic satellite lines depends on the spectral resolution considered, we have estimated the ratios R and G for four specific spectral resolutions (FWHM): RGS-1 at the first order (i.e. $\Delta \lambda=0.073$, 0.075 and 0.078 Å  for Ne IX, Mg XI and Si XIII respectively), LETGS (i.e. $\Delta \lambda=0.05$ Å), HETGS-MEG (i.e. $\Delta \lambda=0.023$ Å), and HETGS-HEG (i.e. $\Delta \lambda=0.012$ Å).

For the n=2, 3, 4 blended dielectronic satellite lines we use the atomic data reported in the Appendix. For the higher-n blended dielectronic satellite lines we use the results from Karim and co-workers. For Z=10 (Ne IX) we use the data from Karim (1993) who gives the intensity factor $F^{*}_{2} \equiv F_2/g_1$ for the strongest ( F^*₂ > 10¹²s^-1) dielectronic satellite lines with n=5-8. For Z=14 (Si XIII), we take the calculations from Karim & Bhalla (1992) who report the intensity factor F^*₂ for the strongest ( F^*₂ > 10¹²s^-1) dielectronic satellite lines with n=5-8. For Z=12 (Mg XI) we have interpolated between the calculations from Karim (1993) for Z=10, and from Karim & Bhalla (1992) for Z=14.

Including the contribution of the blended dielectronic satellite lines, we write for the ratios R and G:

$$% R=\frac{z+satz}{(x+y)+satxy}$$ (31)

$$G=\frac{(z+satz)+((x+y)+satxy)}{(w+satw)},$$ (32)

where satz, satxy and satz are respectively the contribution of blended dielectronic satellite lines to the forbidden line, to the intercombination lines, and to the resonance line, respectively. One can note that at very high density the ³P levels are depleted to the ¹P level, and in that case x+y decreases and R tends to satz/satxy.

At the temperature at which the ion fraction is maximum for the He-like ion (see e.g. Arnaud & Rothenflug 1985; Mazzotta et al. 1998), the differences between the calculations for R (for G) with or without taking into account the blended dielectronic satellite lines are only of about 1$\%$ (9$\%$), $2\%$ (5$\%$), and 5$\%$ (3$\%$) for Ne IX, Mg XI, and Si XIII at the low-density limit and for $T_{\rm rad}=0$K, respectively. On the other hand, for much lower electron temperatures, the effect is bigger since the intensity of the dielectronic satellite lines is proportional to $T_{\rm e}^{-3/2}$. As well, for high values of density ($n_{\rm e}$) at which the intensity of the forbidden line is very weak (i.e. tends to zero), the contribution of the blended dielectronic satellite lines to the forbidden line leads to a ratio R which decreases much slower with $n_{\rm e}$ than in the case where the contribution of the blended dielectronic satellite lines is not taken into account.

  
4.4 Influence of a radiation field

Recently, Kahn et al. (2001) have found with the RGS on XMM-Newton that for $\zeta$ Puppis, the forbidden to intercombination line ratios within the helium-like triplets are abnormally low for N VI, O VII, and Ne IX. While this is sometimes indicative of a high electron density, they have shown that in the case of $\zeta$ Puppis, it is instead caused by the intense radiation field of this star. This constrains the location of the X-ray emitting shocks relative to the star, since the emitting regions should be close enough to the star in order that the UV radiation is not diluted too much. A strong radiation field can mimic a high density if the upper (³S) level of the forbidden line is significantly depopulated via photo-excitation to the upper (³P) levels of the intercombination lines, analogously to the effect of electronic collisional excitation (Fig. 1). The result is an increase of the intercombination lines and a decrease of the forbidden line.

Equation (21) gives the expression for photo-excitation from level m to level p_k in a radiation field with effective blackbody temperature $T_{\rm rad}$ from a hot star underlying the X-ray line emitting plasma. As pointed out by Mewe & Schrijver (1978a) the radiation is diluted by a factor W given by

$$% W=\frac{1}{2}~\left[1-\left(1-\left(\frac{r_{*}}{r}\right)\right)^{1/2}\right],$$ (33)

where r is the distance from the center of the stellar source of radius r_*. Close to the stellar surface the dilution factor $W={1\over 2}$. For stars such as Capella or Procyon, we can take $W={1\over 2}$, because the stellar surface which is the origin of the radiation irradiates coronal structures that are close to the stellar surface (Ness et al. 2001a). In a star such as Algol the radiation originates from another star, and W is much lower (i.e. $W \simeq 0.01$, cf. Ness et al. 2001b), but due to the strong radiation field the radiation effect can still be important.

In their Table 8, Mewe & Schrijver (1978a) give for information the radiation temperature for a solar photospheric field for Z=6, 7, and 8. In Table 3, we report the wavelengths at which the radiation temperature should be estimated for Z=6, 7, 8, 10, 12, 14. These wavelengths correspond to the transitions between the ³S and ³P levels ( $\lambda_{f\to i}$) and the ¹S and ¹P levels ( $\lambda_{6\to r}$).

 

 

Table 3: Wavelengths at which the radiation temperature ( $T_{\rm rad}$) should be determined.

	C V	N VI	O VII	Ne IX	Mg XI	Si XIII
$\lambda_{f\to i}$ (Å)	2280	1906	1637	1270	1033	864
$\lambda_{6\to r}$ (Å)	3542	2904	2454	1860	1475	1200

The photo-excitation from the ³S level and ³P levels is very important for low-Z ions C V, N VI, O VII. For higher-Z ions, this process is only important for very high radiation temperature ($\sim$few 10000 K).

One can note that the photo-excitation between the levels ¹S₀ and ¹P₁ is negligible compared to the photo-excitation between the ³S₁ and ³P_0,1,2 levels. For example, for a very high value of $T_{\rm rad}=30\,000$K the difference between the calculations taken or not taken into account the photo-excitation between ¹S₀ and ¹P₁ is smaller than 20$\%$ for C V, where this effect is expected to be maximum.