2 Model

Consider a fast-moving metal-rich ejecta fragment. Supersonic motion of the fragment (of velocity $v_{\rm k}$, radius R and mass M) in the ambient medium will result in a bow-shock/internal-shock structure creation (Chevalier 1975; Sutherland & Dopita 1995). The bow shock has the velocity $\ga$$v_{\rm k}$ and it is propagating in an ambient medium of the standard cosmic composition. The bow shock creates a hot high pressure gas and also accelerates nonthermal particles. The high pressure gas in the head of the fast moving ejecta fragment could drive an internal shock of velocity $v_{\rm is}$ into the fragment if $v_{\rm is}$exceeds the sound speed in the cold matter:

$$% v_{\rm is} \approx v_{\rm k}~ (\rho_{\rm a}/\rho_{\rm k})^{1/2} \propto v_{\rm k}~ \rho_{\rm a}^{1/2} R^{3/2} M^{-1/2}.$$ (1)

Here $\rho_{\rm a}$ and $\rho_{\rm k}$ are the ambient gas and the dense fragment densities, respectively. We shall also use the density contrast $\chi = \rho_{\rm k}/\rho_{\rm a}$. The characteristic size R of 10¹⁶-10¹⁷ cm is consistent with that resolved by optical observations of Cas A and Puppis A while large clumps were inferred by Chandra observations of Vela shrapnel A by Miyata et al. (2001). For an ejecta fragment of radius $R_{16} \sim$ 3 (where R₁₆ is the radius measured in 10¹⁶ cm) and mass $M \sim 10^{-3}\mbox{$M_{\odot}$ }$ the density ratio $\rho_{\rm a}/\rho_{\rm k} = \chi^{-1}$ is about 4 $\times 10^{-4}$for an ambient medium of number density $\sim$ $0.1 \rm ~cm^{-3}$. However the ratio could be as high as $\ga$0.1 for the ejecta fragments moving through a dense ambient matter of a molecular cloud.

There is a number of factors limiting the fast ejecta fragment lifetime. A fast moving knot is decelerating due to the interaction with the ambient gas. The deceleration time of a knot can be estimated as $\tau_{\rm d} \approx Mv_{\rm k}^2/\rho_{\rm a} v_{\rm k}^3 S \sim \chi R/v_{\rm k}$, where S is the effective crossection of the bow shock. However, the fast knot hydrodynamical crushing effects could have much shorter timescale, providing $\tau_{\rm c} \sim \chi^{1/2} R/v_{\rm k} \sim R/v_{\rm is} \propto M^{1/2} (R \cdot \rho_{\rm a})^{-1/2} v_{\rm k}^{-1}$(e.g. Wang & Chevalier 2001 and references therein). That implies $\tau_{\rm c}/ \tau_{\rm d} \sim \chi^{-1/2} \propto R^{3/2} \rho_{\rm a}^{1/2} M^{-1/2}$. The knot ablation process with surface erosion could also be a limiting factor.

The hydrodynamical estimation of the inner shock velocity $v_{\rm is}$given by Eq. (1) assumes an efficient conversion of the bow shock ram pressure to the knot internal shock. That is a good approximation if the effects of energy loss due to nonthermal particle acceleration and radiative cooling of high pressure gas behind a bow shock are negligible. The latter is the case unless the knot is interacting with dense matter (like molecular cloud clumps).

The Alfvenic Mach number of both the bow shock and the inner shock (if $v_{\rm is}$ is determined by Eq. (1)) is

$$% {\cal M}_{\rm a} = v_{\rm k} (4\pi \rho_{\rm a})^{1/2}/B \approx 460~ v_8~ n_{\rm a}^{1/2}/ ~ B_{-6},$$ (2)

where $n_{\rm a}$ is the ionized ambient gas number density measured in $\rm ~cm^{-3}$, B_-6 is the local magnetic field just before the shock measured in $\mu$G and v₈ is the flow velocity in 10 $^8~\rm ~cm~s^{-1}$. Note here that the Alfvenic Mach number of the inner shock depends on the ambient density but on the magnetic field inside the knot. The sonic Mach number for a shock propagating in a plasma of the standard abundance is

$$% {\cal M}_{\rm s} \approx 85~ v_8~ \cdot [T_4 \cdot (1 + f_{\rm ei})]^{-1/2},$$ (3)

where T₄ is the plasma ion temperature measured in 10⁴ K and $f_{\rm ei} = T_{\rm e}/T_{\rm i}$. In general, for the processes in the precursor and viscous velocity jump of an MHD collisionless shock wave no equilibration between electrons and ions should be assumed (e.g. Raymond 2001). Both cases of $f_{\rm ei} \ll 1$ and $f_{\rm ei} > 1$ could be relevant to the system under consideration. The photoionized gas at the radiative precursor of a fast shock is assumed to have $T_{\rm e} \sim$ (1-2) $\times 10^4$ K and $f_{\rm ei} \ga1$ (e.g. Shull & McKee 1979). On the other hand, plasma heating due to MHD waves dissipation in the vicinity of the viscous subshock heats mostly the ions, providing $f_{\rm ei} < 1$.

2.1 The effect of nonthermal particles on a strong bow shock

For strong collisionless shocks in a magnetized plasma the nonthermal particle acceleration effect is expected to be efficient and a significant fraction of ram pressure is transferred to high energy particles (ions for nonrelativistic shocks). The shock transition of a strong shock of the total Mach number ${\cal M}{\rm t} \gg$ 1 is broadened because of the upstream gas deceleration by nonthermal particle pressure gradient ahead of the viscous gas subshock of a modest Mach number ${\cal M}_{\rm sub} \leq$ 3 (see for a review Blandford & Eichler 1987; Jones & Ellison 1991; Malkov & Drury 2001).

The total compression ratio $R_{\rm t}$ of a strong MHD shock modified by an efficient nonthermal particle acceleration depends on the energy flux carried out by escaping nonthermal particles and the effective adiabatic exponent (e.g. Malkov & Drury 2001). On the other hand the distribution function of nonthermal particles and the bulk flow profile in the shock upstream region are sensitive to the total compression ratio $R_{\rm t}$. Thus, the exact calculation of the escape flux can be performed only in fully nonlinear kinetic simulations. Nevertheless, some approximate iterative approach can be used to make the distribution function consistent with the shock compression. The subshock is the standard gas viscous shock with the compression ratio: $R_{\rm sub} = (\gamma_{\rm g} +1) {\cal M}_{\rm sub}^2$/ $[(\gamma_{\rm g} -1){\cal M}_{\rm sub}^2 + 2]$. The downstream ion temperature T⁽²⁾ behind the modified shock structure can be calculated if $R_{\rm t}$ and $R_{\rm sub}$ are known for the shock of the given velocity and it could be very different from that obtained with the standard single-fluid Rankine-Hugoniot law. The electron temperature just downstream of the shock is expected to be lower than T⁽²⁾.

An exact modeling of the collisionless shock structure with nonthermal particle acceleration effect requires a nonperturbative selfconsistent description of a multi-component and multi-scale system including strong MHD turbulence dynamics. That modeling is not feasible at the moment. We shall use some simplified description of a strong shock with some apropriate parameterization of governing parameters which are: (a) the nonthermal particle diffusion coefficients; (b) the ion injection rate; (c) the maximum momentum of accelerated particles. The gas heating mechanism due to MHD waves dissipation in the shock precursor must also be specified in that simplified approach, providing the connection between the total compression and the gas subshock Mach number ${\cal M}_{\rm sub}$. Assuming that the main heating mechanism of the gas in the precursor region is due to Alfven waves dissipation, Berezhko & Ellison (1999) obtained a simple relation for the total compression of the shock $R_{\rm t} \approx 1.5 {\cal M}_{\rm a}^{3/8}$, while the subshock had ${\cal M}_{\rm sub} \sim 3$. The relation is valid under the condition of ${\cal M}_{\rm s}^2 \gg {\cal M}_{\rm a}$ in the far upstream flow. They also noted that the subshock compression ratio is not too sensitive to the ion injection rate if the rate exceeds $5 \times 10^{-5}$ to provide the shock modification by the accelerated ions. We obtain then the ion temperature just behind the modified strong shock (measured in 10⁶ K)

$$% T^{(2)}_6 \approx 0.32 \cdot \phi({\cal M}_{\rm sub}) \cdot... ... k8}^{5/4} n_{\rm a}^{-3/8} B_{-6}^{3/4}(1 + f_{\rm ei})^{-1},$$ (4)

where $\phi({\cal M}_{\rm sub}) = [2 \gamma_{\rm g} {\cal M}_{\rm sub}^2 - (\gamma_{\rm g} -1)] /[(\gamma_{\rm g} -1){\cal M}_{\rm sub}^2 + 2]$, and $\phi(3) \approx 3.7$ for $\gamma_{\rm g}$= 5/3. Then the electron and ion temperatures equilibrate in the postshock region. The initial electron temperature in the shock downstream depends also on the collisionless electron heating (see e.g. Bykov & Uvarov 1999; Laming 2001). The turbulent gas heating due to acoustic instability wave dissipation could be more efficient however, if the acoustic instability develops (Malkov & Drury 2001).

The maximal momenta of the accelerated ions are limited by MHD wave damping due to ion-neutral collisions in the case of a fragment interacting with dense molecular cloud. The effect was accounted for following the work by Drury et al. (1996). The maximal momenta of the ions are $\la$10² GeV/c in that case. In the case of a fragment propagating through a tenuous medium the maximal momentum is limited by the finite scale of the strong scattering region with typical values of the maximal momentum $\gg$10² GeV/c.

It should be noted that we still have no direct observational evidence for the cosmic ray ions accelerated in SNR shocks. At the same time there is direct evidence for ion and electron acceleration by strong cosmic-ray modified shocks in the interplanetary medium (e.g. Terasawa et al. 1999). In the interplanetary shock of velocity $\sim$375 $\rm ~km~s^{-1}$(in the solar wind frame) studied by Terasawa et al. (1999) with the GEOTAIL satellite the partial pressure of accelerated nonthermal particles was found to be in a rough balance with the thermal proton pressure in the shock downstream. That implies a high efficiency of nonthermal ion production. The diffusion coefficients for both electrons and ions were estimated to be $\sim$10 $^{18}~\rm ~cm^2~s^{-1}$ in the shock vicinity that is at least two order of magnitude smaller than the typical interplanetary values (Shimada et al. 1999). For SNR shocks Dorfi (2000) estimated the ion acceleration efficiency to be 0.24 for some reasonable parameters.

The observed synchrotron emission from SN shells has been considered for a long time as direct evidence for relativistic electron acceleration. The efficiency of transformation of the kinetic energy of a nonrelativistic shock bulk flow into the nonthermal electron population is not expected to be very high, being below a few percent. However even a percent range efficiency of the nonthermal electron acceleration would provide an observable effect of K $_{\rm\alpha}$ photon emission from metallic knots irradiated by high energy electrons. Bulk heating and ionization of the knot interior due to deep penetration of energetic ions and electrons could also be important for enhanced ablation.

2.2 Nonthermal electron acceleration

To simulate the spectra of nonthermal electrons in the fragment body we solve the kinetic equation for the distribution function of the electrons, accounting for injection, diffusive transport, advection, first and second order Fermi acceleration and synchrotron and the Coulomb losses. Electron kinetics in supercritical collisionless shocks was modeled by Bykov & Uvarov (1999). They showed that strong MHD fluctuations generated by kinetic instabilities of ions are responsible for heating and pre-acceleration of nonthermal electrons on a very fine scale, of the order of several hundreds of inertial lengths, in the vicinity of the viscous jump (subshock) of a collisionless shock. We have described the details of the code, diffusion model and loss functions used elsewhere (Bykov & Uvarov 1999; Bykov et al. 2000).

There are three zones in the one-dimensional model: the pre-shock region (I), the shock transition region (II), and the post-shock flow (III), from where nonthermal emission from shock-accelerated particles originates. In order to calculate the spectra of nonthermal electrons in these regions, we used the following kinetic equation for the nearly-isotropic distribution function N_i(z,p) (i = I - III):

$$k_i(p) \: \frac{ \partial^2 N_i(z,p) }{ \partial z^2 } + \frac{1}... ...l}{\partial p}\: N_i\: \left[\frac{ \partial}{ \partial z }\: u_i \right] = 0 .$$ (5)

This Fokker-Planck-type equation takes into account diffusion and advection [bulk velocity u_i(z)] of electrons in phase space due to interactions with MHD waves and the large-scale MHD flow. Here z is the coordinate along the shock normal. L_i(p) is the momentum loss rate of an electron due to Coulomb collisions in a partially ionized plasma and synchrotron/inverse Compton radiation. The momentum diffusion coefficient D(p) is responsible for the second order Fermi acceleration, and k_i(p) is the fast particle spatial diffusion coefficient. For low-energy electrons the Coulomb and ionization losses are important everywhere except for the narrow shock transition region (II), where acceleration is fast enough to overcome the losses and where nonthermal electron injection occurs.

To accelerate the electrons injected in the shock transition region to relativistic energies MHD turbulence should fill the acceleration region. As we know from interplanetary shock observations, turbulence generation is a generic property of collisionless MHD shocks. We used the parameterization of the momentum dependent diffusion coefficient k_i(p)of an electron of momentum p, described in details by Bykov et al. (2000). The diffusion coefficients are momentum dependent with $k_i(p) \propto k_0 \cdot v p$above 1 keV. The normalization value k₀ is the electron diffusion coefficient at 1 keV.

The electron temperature in the (photo)ionized shock precursor was fixed to $T_{\rm e} = 2 \times 10^4$ K for the fragments propagating in the dense clouds of the gas of number density $\gg$1  $\rm ~cm^{-3}$. The fragment velocity should exceed $\sim$ $250 \rm ~km~s^{-1}$ to provide enough UV radiation to complete H and He (to He⁺) ionization in the precursor of the bow shock, if it is modified by the nonthermal particle pressure. The velocity is higher than the value of $\sim$ $110 \rm ~km~s^{-1}$ derived by Shull & Mckee (1979), because of the weaker gas heating by the modified MHD shock structure with a smooth precursor and a gas subshock of ${\cal M}_{\rm sub} \sim 3$. It should be noted however that even at lower shock velocities where the upstream photoionization is inefficient, the preshock ionization could be provided by the fast particles accelerated at the collisionless shock. For the fragments propagating in hot tenuous plasma the electron temperature was assumed to be equal to the plasma temperature (i.e. $f_{\rm ei}$ = 1).

The energy spectrum of the nonthermal electrons in the bow shock of a metal-rich ejecta fragment is shaped by the joint action of the first and second order Fermi acceleration in a turbulent plasma with substantial Coulomb losses. The electron spectra are calculated for different values of the bow shock velocity assuming that the diffusion coefficient is parameterized as in Bykov et al. (2000) with different values of k₀ for 1 keV electron for different environments. The minimal value of the diffusion coefficient of an electron is $k_0^{{\rm min}} \approx 6.6\times 10^{16} B^{-1}_{\mu{\rm G}} (E/{\rm keV}) \rm ~cm^2~s^{-1}$. Note that the minimal value is only achievable if strong fluctuations of the magnetic field are present at the electron gyro-radii scale. For slightly superthermal electrons the fluctuations cannot be generated by the ions. Whistler type fluctuations could provide the required electron scale fluctuations (e.g. Levinson 1996). However in the case of a strong MHD turbulence the electron transport by the vortex type fluctuations of ion gyro-radii scale would provide an effective diffusion coefficient much larger than $k_0^{{\rm min}}$ (Bykov & Uvarov 1999). We consider here rather a conservative case of that large diffusion coefficient.

The electron acceleration model described above is one-dimensional. The transverse gas flow of velocity $u_{\perp}$behind the bow shock front would affect the nonthermal particle distribution. The exact 2-D and 3-D modeling of the postshock flow with account of shock modification effect is not feasible at the moment. However, the transverse flow effect on high energy electrons is only marginal if the particle diffusion time across the transverse flow is shorter than the transverse advection time scale i.e. $\delta_{\perp}/R \cdot u_{\perp} \delta_{\perp}/k(p) <$ 1. Here $\delta_{\perp}$ is the width of the transverse gas flow behind the bow shock (typically it is a relatively small fraction of the fragment radius R). In our case the condition is satisfied for 100 keV - MeV regime electrons dominating the $\rm K_\alpha$ lines production in the ejecta fragment body. We account for the presence of a postshock cooling layer of the shocked ambient matter situated in front of the metal rich fragment. The curves 1-3 in Fig. 1 illustrate the effect of Coulomb losses on low energy nonthermal electrons diffusing through the postshock layer.

Figure 1: The nonthermal electron distribution function ( ${\tilde N(p,x)} = N(p,x)p^2p_{\rm T}$) as a function of dimensionless particle momentum $p/p_{\rm T}$($p_{\rm T}$ is the momentum of a thermal electron in the upstream region). We show the distributions at different distances from the bow shock plane for different ambient media: a) a fragment of velocity $v_{\rm k} = 3200 \rm ~km~s^{-1}$in a tenuous medium; b) an SN fragment of velocity $v_{\rm k} = 1080 \rm ~km~s^{-1}$ in a molecular cloud (see Sect. 2.3 for details). The depths from the bow shock plane are 106, 1012, 1015, 3$\times$ 1016 cm for the curves 1-4, respectively.

2.3 Ambient conditions and the fragment state

As a generic example we consider a fast moving fragment of $v_{\rm k} \sim$ 1000 $\rm ~km~s^{-1}$, mass $M = 10^{-3} ~\mbox{$M_{\odot}$ }$ and radius R₁₆ = 3 dominated by oxygen and containing about $\sim$10 $^{-4}~\mbox{$M_{\odot}$ }$ of any metal impurities like Si, S, Ar, Ca, Fe. That fragment we refer to as "the standard knot''.

The electron distribution was simulated here for two distinct representative cases of the ambient environment. The first one is an SN fragment propagating through a dense ambient medium of number density $\sim$10 $^3~\rm ~cm^{-3}$, relevant to an SN explosion in the vicinity of a molecular cloud. If the ambient magnetic field follows the scaling $B_{\rm a} \propto n^{1/2}$ (e.g. Hollenbach & McKee 1989) one may expect the magnetic field values of $\ga$100 $\mu$G in the dense ambient medium (cf. Blitz et al. 1993). The second case is a SN fragment propagating through a tenuous medium of number density $\sim$0.1 $~\rm ~cm^{-3}$, temperature T $\ga$ 10⁴ K and magnetic field of $\mu$G range. Correspondingly, the diffusion coefficient normalization factors for keV regime electrons were $k_0 \sim 10^{18} \rm ~cm^2~s^{-1}$ in the dense medium case and $k_0 \la10^{20} \rm ~cm^2~s^{-1}$ for the tenuous medium. The gas ionization state in the bow shock downstream was simulated here as described in the paper of Bykov et al. (2000).

The gas ionization state inside an oxygen rich fragment body was estimated following the model by Borkowski & Shull (1990). They studied the structure of a steady state radiative shock (inner shock) of velocities $v_{\rm is}$from 100 to 170 $\rm ~km~s^{-1}$ (cf. Eq. (1)) in a pure oxygen gas with electron thermal conduction. The postshock temperature behind the 140 $\rm ~km~s^{-1}$ shock with thermal conduction was found to be 270 000 K in the immediately postshock region and dropped to roughly 4700 K after the oxygen depth about $2 \times 10^{15} \rm ~cm^{-2}$ in the downstream photoionization zone. The dominant ionization state of oxygen in the shock radiative precursor and in the immediate postshock region was OIII while it was OII in the downstream photoionization zone. Nonthermal electrons could also be an important factor in modeling the dominant ionization state of the ions in the fragment body (Porquet et al. 2001).

In Fig. 1 the nonthermal parts of the electron distribution functions ${\tilde N(p,x)} = N(p,x)p^2p_{\rm T}$ are shown at the different depths of an SN fragment. Here $p_{\rm T}$ is the momentum of a thermal electron in the far upstream region. Note that the upstream gas electron temperatures are different for the two ambient media we displayed in Fig. 1. In Fig. 1 we showed only nonthermal parts of the electron distributions because the electron temperatures in both presented cases are below those required for efficient $\rm K_\alpha$ line production in the ejecta fragments. The threshold for K-shell ionization of Si is 1.8 keV (i.e. $p/p_{\rm T} >$ 33 in Fig. 1) while that for Fe is about 7.1 keV (i.e. $p/p_{\rm T} >$ 64 in Fig. 1). Fast electrons accelerated in the MHD collisionless shock diffuse through the postshock layer and cold metallic knot suffering Coulomb losses as is clear from the Fig. 1. They produce the $\rm K_\alpha$ lines in the fragment body due to radiative transitions following the removal of the 1s atomic electrons.

2.4 K-shell ionization and X-ray line emissivity

To calculate the production rate of characteristic X-ray line emission from the metal rich fragment irradiated by an intense flux of energetic electrons accelerated by an MHD shock wave the atomic inner shell ionization cross sections and the fluorescent yields are required. The experimental electron-impact K-shell ionization cross-sections and the fluorescent yields were compiled recently by Liu et al. (2000). The standard Bethe theory of inner shell ionization (e.g. Powell 1976) was used to fit the data for nonrelativistic energies of ionizing electrons, from the threshold up to 50 keV and the relativistic formulae by Scofield (1978) above 50 keV.

We fit the K-shell ionization cross section following the parameterization given by Scofield (1978)

$$% \sigma_i = \frac{A}{\beta^2} \cdot (b + b_1 + b_2/\epsilon + b_3/p^2 + b_4\cdot b/p + b_5/p^4).$$ (6)

Here the cross-section $\sigma_i$ is given as a function of the incident electron momentum p(measured in $m_{\rm e} c$), b = ln $({\rm p}^2) - \beta^2$, $\beta = v/c$. The fitting parameters A, b₁ - b₅ were calculated by Scofield (1978) for the ions of $Z \geq$ 18. We corrected the fitting parameters to account for the present laboratory measurements compiled by Liu et al. (2000).

The logarithmic asymptotic of the crossection at relativistic energies, given by Eq. (6) is important for our modeling. One can see in Fig. 1 (curves 3 and 4) that the electron spectra have rather flat maxima at mildly relativistic energies ( $p/p_{\rm T} \ga400$) in the most of the volume of a dense metal-rich fragment. Thus, the K-shell ionizations by relativistic electrons are substantial.

The radiative decay of K-shell vacancy induced by fast electron results in ${\rm K}_{\rm\alpha}$ $\rm (2p \rightarrow 1s)$ and $\rm K_{\beta}$ $\rm (3p \rightarrow 1s)$ lines production. The $I({\rm K}_{\rm\beta})/I({\rm K}_{\rm\alpha})$ intensity ratio is about 0.14 (e.g. Scofield 1974) for FeI-FeIX and it is decreasing for higher iron ionization stages (e.g. Jakobs & Rozsnyai 1986). The ratio is an increasing function of the atomic number Z. Note however, that the optical depths for ${\rm K}_{\rm\beta}$ and ${\rm K}_{\rm\alpha}$ lines could be different, especially for highly ionized iron, and that would affect the observed line ratio. Thus, observing the ratio $I({\rm K}_{\rm\beta})/I({\rm K}_{\rm\alpha})$ one may constrain the source optical depth. Both $I({\rm K}_{\rm\alpha})$ and $I({\rm K}_{\rm\beta})$ Fe lines were detected recently with Chandra from Sgr B2 giant molecular cloud by Murakami et al. (2001). The observed ratio is somewhat below 0.2 indicating some possible optical depth effect, but it is still marginally consistent with the value 0.14 expected for a transparent system.

It is important to note here that shock-accelerated energetic nuclei can also provide efficient K-shell ionization. The emission spectra from the decay of inner-shell ionizations produced by a collision with an MeV-regime ion have multiple satellites (peaks) at energies higher than that of the electron-induced ${\rm K}_{\rm\alpha}$ line (e.g. Garcia et al. 1973). We shall discuss the effect of ions in detail elsewhere.

2.5 The optical depth effect

The maximal column density of a metal (of atomic weight A) in a spherical ejecta fragment $N^{(A)}_{\rm max} = 5.6\times 10^{22}~(M^{(A)}/\mbox{$M_{\odot}$ })~A^{-1}~R_{17}^{-2}~~ {\rm cm}^{-2}$can be high enough to provide optical depths $\tau \ga1$. We use below $N^{(A)} = (1/2) N^{(A)}_{\rm max}$as the resonant scattering column density in the model assuming the shock situated at the fragment center. The resonant line scattering effect (see, for the review, Mewe 1990) can be important for the metal rich SN fragments. The optical depth at the line center of an ion due to the resonant line scattering was given by Kaastra & Mewe (1995) and can be expressed as:

$$% \tau \approx 1.34~f_{\rm abs}~N_{15}^{(A)} E_{\rm ph}^{-1} (A/T)^{1/2} (1 + 0.5 w_6^2 A/T)^{-1/2}.$$ (7)

where $E_{\rm ph}$ (in keV) is the line photon energy, $f_{\rm abs}$ is the absorption oscillator strength, T is the ion temperature (in eV), N₁₅^(A) is the ion column density (in 10¹⁵ cm^-2), w₆ is the gas micro-turbulence velocity (in 10 $\rm ~km~s^{-1}$).

To describe the radiative transfer for the line radiation produced inside the absorbing medium we used a simplified escape probability approach. The mean escape probability $p_{\rm f}(\tau)$ can be approximated as $p_{\rm f}(\tau) \approx [1 + a \tau]^{-1}$, where $a \approx 0.43$ for $\tau \la50$ (Kaastra & Mewe 1995). The photon absorption due to the resonant scattering of the K-shell line is most important for the ions with an incomplete L-shell (e.g. OI, Si VI, Ar X, Fe XVIII) and relatively low fluorescent yields. The resonant absorption is not effective for the neutral or the low ionization stages of Fe, Ca, Ar, Si with completed L-shells where we applied the photoabsorption by Morrison & McCammon (1983).