1 Introduction

The new X-ray satellites (Chandra, XMM and Astro-E) will offer unprecedented high spectral resolution and high sensitivity spectra. Indeed, it will be possible to observe and to separate, in the X-ray range, the three most intense lines of He-like ions: the resonance line (w: 1s^{2 1}S₀ - 1s2p ¹P ₁), the intercombination lines (x,y: 1s^{2 1}S₀ - 1s2p ³P_2,1 respectively) and the forbidden line (z: 1s^{2 1}S₀ - 1s2s ³S₁). They correspond to transitions between the n=2 shell and the n=1 ground state shell (see Fig. 1).

The ratios of these lines are already widely used for collisional (coronal) plasma diagnostics of various types of objects: solar flares, supernovae remnants, the interstellar medium and tokamak plasmas, i.e. for very hot collisional plasmas (Mewe & Schrijver [1978a], [1978b]; Winkler et al. [1981]; Doyle & Schwob [1982]; and Pradhan & Shull [1981]). As shown by Gabriel & Jordan ([1969], [1972], [1973]), these ratios are sensitive to electron density ( R(n_e), Eq. (1)) and to electronic temperature ( G(T_e), Eq. (2)):

$${ R~(n_{\rm e})~=~\frac{z}{(x+y)}}$$ (1)

$${ G~(T_{\rm e})=\frac{z+(x+y)}{w}}\cdot$$ (2)

As emphasized by Pradhan ([1985]), Liedahl ([1999]) and Mewe ([1999]) (see also Paerels et al. [1998]), these plasma diagnostics could be also extended to study photoionized plasmas. Indeed, Pradhan has calculated the R and G ratios for highly charged ions (Ar XVII and Fe XXV) in "recombination dominated non-coronal plasmas''. We present numerical calculations of these ratios, for six lighter ions, which could be applied directly for the first time to Chandra and XMM observations of the Warm Absorber present in Active Galactic Nuclei (AGN), and especially in Seyfert1.

The Warm Absorber (WA) is a totally or a partially photoionized medium (with or without an additional ionization process), first proposed by Halpern ([1984]) in order to explain the shape of the X-ray spectrum of the QSO MR2251-178, observed with the Einstein Observatory. Its main signatures are the two high-ionization oxygen absorption edges, O VII and O VIII at 0.74 keV and 0.87 keV respectively, seen in fifty percent of Seyfert 1 galaxies at least (Nandra $\&$ Pounds [1994]; Reynolds [1997]; George et al. [1998]). According to Netzer ([1993]), an emission line spectrum from the WA should also be observed. Indeed, He-like ion lines have been observed in different types of Seyfert galaxies (NGC 3783: George et al. [1995], MCG-6-30-15: Otani et al. [1996], E 1615+061: Piro et al. [1997], NGC 4151: Leighly et al. [1997], NGC 1068: Ueno et al. [1994], Netzer & Turner [1997], and Iwasawa et al. [1997]). The WA is supposed to be at least a two-zone medium with an inner part (called the "inner WA'') associated with O VIII and an outer part (called the "outer WA''), less ionized, associated with O VII (Reynolds [1997]; Porquet et al. [1999]). Furthermore, the O VII line is predicted to be the strongest line associated with the outer WA; the Ne IX line is predicted to be one of the strongest lines formed in the inner WA (Porquet et al. [1998]).

Figure 1: Simplified Gotrian diagram for He-like ions. w, x, y and z correspond respectively to the resonance, intercombination and forbidden lines. Full curves: collisional excitation transitions, broken curves: radiative transitions and thick dot-dashed curves: recombination (radiative and dielectronic). Note: the broken arrow (1S0 to the ground level) correspond to the 2-photon continuum

The ionization processes, that occur in the Warm Absorber, are still not very well known. Indeed, even though the WA is commonly thought to be a photoionized gas, an additional ionization process cannot be ruled out (Porquet & Dumont [1998]; Porquet et al. [1999]; Nicastro et al. [1999]). Thus, in the present paper, we do not restrict ourselves to only a single type of plasma, but rather study the following cases.

We consider a "pure photoionized plasma'' to be a plasma ionized by high energy photons (external ionizing source). For such a plasma, H-like radiative recombination (and dielectronic recombination at high temperature) are dominant compared to electronic excitation from the ground level (1s²) of He-like ions. The lines are formed by recombination. A "hybrid plasma'' is a partially photoionized plasma, but with an additional ionization process, e.g. collisional (internal ionizing source). For this case, He-like electronic excitation processes from the ground level are usually as important as H-like recombinations, and may even dominate. The lines are formed by collisional excitation from the ground level with or without recombination. In the next section, we introduce the atomic data calculations needed for such plasmas and we emphasize the role of upper-level radiative cascade contributions calculated in this paper for the populations of the n=2 shell levels. In Sect. 3, we develop line diagnostics of the ionization process (temperature) and the density for pure photoionized and hybrid plasmas. We give the corresponding numerical calculations of the line ratios for C V, N VI, O VII, Ne IX, Mg XI, and Si XIII. In Sect. 4, we give a practical method for using these results to determine the physical parameters of the WA, in the context of the expected data from the new X-ray satellites (Sect. 5).