1 Introduction

The new generation of X-ray satellites (Chandra, XMM-Newton) enables us to obtain unprecedented high spectral resolution and high S/N spectra. The wavelength ranges of the RGS (6-35 Å), of the LETGS (2-175 Å), and of the HETGS (MEG range: 2.5-31 Å; HEG range: 1.2-15 Å) contain the helium-like line "triplets'' from C V (or N VI for the RGS, and for the HETGS-HEG) to Si XIII. The triplet consists of three close lines: the resonance line, the intercombination line and the forbidden line. The helium-like triplets provide electron density ( $n_{\rm e} \sim 10^8{-}10^{13}$ cm^-3) as well as electron temperature ( $T \sim 1{-}10$ MK) as first shown by Gabriel & Jordan (1969). The line ratios of these He-like triplets enable also the determination of the ionization processes (photo-ionization and/or collisional ionization) which prevail in the plasma (Porquet & Dubau 2000; Liedahl 1999).

The ratios of these lines are already widely used for collisional solar plasma diagnostics (e.g., Gabriel & Jordan 1969; Doyle 1980; Keenan et al. 1987; McKenzie & Landecker 1982).

Recently, also theoretical calculations for photo-ionized plasmas or "hybrid'' plasmas (photo-ionization plus collisional ionization) have been given by Porquet & Dubau (2000) (hereafter referred to as Paper I). Their calculations have been already applied to spectra of Seyfert galaxies (e.g. NGC5548, Kaastra et al. 2000; Mkn3, Sako et al. 2000; NGC4151, Ogle et al. 2000; NGC4051, Collinge et al. 2001, etc.).

We present here calculations of these ratios, from C V to Si XIII, which could be applied directly for the first time to Chandra and XMM-Newton observations of extra-solar collisional plasmas such as stellar coronae. These calculations have been done to apply an improved model to the density analysis of the RGS, the LETGS and the HETGS spectra of various late-type stars such as Capella, Procyon, and $\alpha$ Centauri (e.g., Audard et al. 2001; Ness et al. 2001a; Mewe et al. 2001) and also to O stars such as $\zeta$ Puppis (Kahn et al. 2001). Our model is to be considered as an improvement of various previous calculations for solar plasmas such as done by e.g., Gabriel & Jordan (1969), Blumenthal et al. (1972), Mewe (1972), Mewe & Schrijver (1978a, 1978b, 1978c), Pradhan & Shull (1981), Mewe & Gronenschild (1981), Mewe et al. (1985), and Pradhan (1982, 1985). The calculations are partly based on recent work by Porquet & Dubau (2000).

In the next three sections, we introduce the plasma diagnostics and the atomic processes and atomic data taken into account in the calculations. In Sect. 5, we display the results for R and G calculated for four specific spectral resolutions (Full Width at Half Maximum: FWHM): RGS, LETGS, HETGS-MEG, HETGS-HEG, over a broad range of physical parameters: $n_{\rm e}$, $T_{\rm e}$, radiation temperature ( $T_{\rm rad}$), and radiation dilution factor (W).