Nonequilibrium ionization states in galaxy clusters
D. A. Prokhorov^{1,2,3}
1  UPMC Université Paris 06, UMR 7095, Institut
d'Astrophysique de Paris, 75014 Paris, France
2  CNRS,
UMR 7095, Institut d'Astrophysique de Paris, 75014 Paris, France
3  Korea Astronomy and Space Science Institute, 611 Hwaamdong,
Yuseonggu, Daejeon 305348, Korea
Received 28 August 2009 / Accepted 28 October 2009
Abstract
Context. Xray imaging observatories have revealed hydrodynamic structures with linear scales of 10 kpc
in clusters of galaxies, such as shock waves in the 1E065756 and A520
galaxy clusters and the hot plasma bubble in the MKW 3s cluster. The
future Xray observatory IXO will for the first time resolve the metal
distribution in galaxy clusters at the these scales.
Aims. Heating of plasmas by shocks and AGN activities can result
in nonequilibrium ionization states of metal ions. We study the effect
of the nonequilibrium ionization at linear scales of 50 kpc in galaxy clusters.
Methods. A condition for nonequilibrium ionization is derived
by comparing the ionization timescale with the age of hydrodynamic
structures. Modeling of nonequilibrium ionization is performed at a
point in time when the plasma temperature suddenly changes. An analysis
of the relaxation processes of the FeXXV and FeXXVI ions by means of
eigenvectors of the transition matrix is given.
Results. We conclude that the nonequilibrium ionization of iron can occur in galaxy clusters if the baryonic overdensity
is smaller than ,
where
is the ratio of the hydrodynamic structure age to the Hubble time. Our
modeling indicates that the emissivity in the heliumlike emission
lines of iron increases as a result of the deviation from the
ionization equilibrium. A slow process of heliumlike ionic fraction
relaxation was analyzed. A new way to determine a shock velocity is
proposed.
Key words: galaxies: clusters: general  atomic processes  shock waves
1 Introduction
Clusters of galaxies are gravitationally bound structures of a mass of and a size of 13 Mpc (for a review, see Kaastra et al. 2008). Their mass budget consists of dark matter (80%), hot diffuse intracluster plasma (20%) and a small fraction of other components such as stars and dust. The mean baryonic overdensity in galaxy clusters equals , where and are the mean hydrogen densities in galaxy clusters and in the Universe, respectively.
Many chemical elements reside in galaxy clusters. The plasma temperatures keV in galaxy clusters are close to the values of the Kshell ionization potentials of heavy elements ( I_{Z}=Z^{2} Ry, where Z is the atomic number and Ry the Rydberg constant). Emission lines from heavy elements were detected by Xray telescopes in galaxy clusters. The current instruments (XMMNewton, Chandra, and Suzaku) have largely enhanced our knowledge on the chemical abundances of many elements. The metal abundances of around 0.3 Solar Units in Anders & Grevesse (1989) were derived under the assumptions of a collisional ionization equilibrium (for a review, see Werner et al. 2008).
Nonequilibrium processes such as nonequilibrium ionization and relaxation of the ion and electron temperatures are usually taken into account only in the outskirts of galaxy clusters and in the warmhot intergalactic medium (WHIM) where the baryonic overdensity is less than 200 (e.g. Yoshikawa & Sasaki 2006; Prokhorov 2008). However, we show that nonequilibrium ionization can also be produced as the result of merging processes and AGN activity in galaxy clusters where the baryonic overdensity .
Evidences for merging processes of galaxy clusters and AGN activity, such as shocks and hot plasma bubbles, were revealed by means of Chandra highresolution observations. For example, strong shocks in the 1E065756 and A520 galaxy clusters propagating with a velocity of 4700 km s^{1} and 2300 km s^{1}, respectively, were derived by Markevitch et al. (2002) and Markevitch et al. (2005). The corresponding Mach numbers of the shocks are 3.0 and 2.1. Hot plasmas inside bubbles arising from AGN activity were detected in galaxy clusters (e.g. Mazzotta et al. 2002). Heating of plasmas produced by shocks or AGNs can result in a nonequilibrium ionization state.
We study the effect of nonequilibrium ionization near merger shock fronts and in hot plasma bubbles. We give a theoretical analysis of collisional nonequilibrium ionization in Sect. 2. We show the importance of this effect in galaxy clusters numerically in Sect. 3. We analyze heliumlike and hydrogenlike nonequilibrium ionization states by means of eigenvectors of the transition matrix in Sect. 4. A new approach to determine the value of the shock velocity is considered in Sect. 5, and our results are discussed in Sect. 6.
2 A condition for nonequilibrium ionization
Nonequilibrium ionization is often assumed in supernova remnants (e.g. Gronenschild & Mewe 1982; Masai 1994) and may be important in the WHIM (Yoshikawa & Sasaki 2006). The nonequilibrium ionization state in the linked region between the Abell 399 and Abell 401 clusters was also studied by Akahori & Yoshikawa (2008). We are going to show that nonequilibrium ionization can occur not only in the outskirts of galaxy clusters but also inside galaxy clusters, in which merging processes and AGN activity play a role. We derive here a condition on the baryonic overdensity for a deviation from collisional ionization equilibrium, by comparing the ionization timescale with the age of hydrodynamic structures.
The number of collisions between electrons and an ion resulting in
electron impact ionization per unit time is
,
where
is the characteristic value of the
ionization crosssection,
corresponds to the
threshold velocity (energy) of the ionization process, and
is the number density of electrons which have
sufficient energy for electron impact ionization. The ionization
timescale is given by
,
and therefore
As was noted by Yoshikawa & Sasaki (2006) and Akahori & Yoshikawa (2008), heliumlike and hydrogenlike ions are interesting for the analysis of nonequilibrium ionization. Therefore, electron impact ionization of a heliumlike ion will be considered in this section as a physically important case (a consideration of hydrogenlike ions is analogous).
For heliumlike ions the characteristic value of the ionization
crosssection is approximately (see Bazylev & Chibisov 1981)
where is the Bohr radius, Z is the atomic number.
Since the ionization potential of a Helike ion is approximately
^{}, the electron
threshold velocity
can be approximated as
The number density of electrons with energies higher than the ionization potential of a Helike ion is
where n_{0} is the plasma number density, is the dimensionless threshold momentum and is the Maxwellian distribution.
If the dimensionless threshold momentum
,
a simplified form of Eq. (4) is given by
Using Eqs. (2), (3) and (5) we rewrite Eq. (1) as
(6) 
It is most convenient to write the plasma number density in terms of the baryonic overdensity , where the critical density is , and to denote the ratio of the thermal energy kT and the ionization potential by . Thus,
(7) 
where .
Ionization states will be nonequilibrium if the ionization
timescale
is longer than the hydrodynamic
structure age
,
i.e.
.
This condition is equivalent
to the inequality
In an important case of the iron ions (Z = 26), the numerical value of the first dimensionless term on the righthand side of Eq. (8) is
(9) 
and, therefore,
(10) 
In rich galaxy clusters with a plasma temperature of keV, the value of the function is 4.6. Therefore we find in this case and conclude that, if the hydrodynamic structure age is on the order of 10^{7}10^{8}years (i.e. lies in the range ), then nonequilibrium ionization occurs in galaxy clusters where the baryonic overdensity (see Eq. (8)).
Mazzotta et al. (2002) have estimated the age of a hot plasma bubble of a diameter of 50 kpc to be yr, which is much shorter than the age of the MKW 3s cluster. In the 1E065756 and A520 clusters the downstream velocities of the shocked gas flowing away from the shock are 1600 km s^{1} and 1000 km s^{1} (Markevitch et al. 2002, 2005), therefore the shocked gas covers a distance of 50 kpc in and yrs respectively. Thus, in light of the above conclusion nonequilibrium ionization can occur at linear scales of 50 kpc in galaxy clusters in which merging processes and AGN activity is present.
3 Modeling of nonequilibrium ionization
Nonequilibrium ionization occurs when the physical conditions of the plasma, such as the temperature, suddenly change. Shocks, for example, can lead to an almost instantaneous rise in temperature and to a deviation from the ionization equilibrium. However, it takes some time for the plasma to respond to an instantaneous temperature change, as the ionization balance is recovered by collisions.
In this section we consider the following situation: the plasma temperature instantaneously increases from kT_{1}=3.4 keV to kT_{2}=10.0 keV. Such a temperature change may correspond to a temperature jump at a shock with a Mach number M=2.6 or to plasma heating by AGN activity. We assume that the age of the hot plasma region is yr and the baryonic overdensity is , which corresponds to the plasma number density in the postshock region in the A520 cluster (see Fig. 2b of Markevitch et al. 2005). Following Markevitch (2006) we assume that the electron and ion temperatures are equal.
Figure 1: Dependence of the Helike (solid line) and Hlike (dashed line) ionic fractions of iron on the dimensionless time yr). 

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At the temperature
kT_{1}=3.4 keV the ionic fractions of Lilike,
Helike and Hlike ions of Fe relative to the total Fe abundance are
,
,
and
respectively.
Therefore we consider below four ironionstates Fe(+23), Fe(+24),
Fe(+25) and Fe(+26). In that case the collisional ionization rate
equation for each element is written as
where is the vector with four components (n_{+23}, n_{+24}, n_{+25}, n_{+26}), normalized in a way that , which corresponds to the four ironionstates mentioned above, I_{+ z} and R_{+ z} represent the rate coefficients for ionization and recombination from an ion of charge z to charges z+1 and z1, respectively. All the coefficients necessary to calculate the direct ionization cross sections are taken from Arnaud & Rothenflug (1985), the radiative recombination rates are taken from Verner & Ferland (1996), and the dielectronic recombination rates are taken from Mazzotta et al. (1998). To solve the system of Eq. (11), we use the fourth order RungeKutta method.
The timedependence of the Helike and Hlike ionic fractions of iron is shown in Fig. 1.
At the temperature kT_{2}=10.0 keV the equilibrium ionic fractions of Helike and Hlike ions of iron are and respectively. Therefore, the Helike ionic fraction, which equals 35% at a time of yr, does not reach its equilibrium value and nonequilibrium ionization occurs. However, the Hlike ionic fraction almost reaches its equilibrium value at a time of yr.
We now show that the effect of nonequilibrium ionization on the heliumlike emission lines of iron can be significant and that nonequilibrium ionization leads to the increase of volume emissivity in the heliumlike spectral lines.
The heliumlike volume emissivity for a chemical element of atomic
number Z is given by
(12) 
where is the electron number density, is the hydrogen number density, A_{Z} is the abundance of the considered chemical element, n_{+( Z2)}and n_{+( Z1)} are the ionic fractions of heliumlike and hydrogenlike ions respectively, Q_{ +(Z2)} is the impact excitation rate coefficient and is the rate coefficient for the contribution from radiative recombination to the spectral lines. Excitation rate coefficients are taken from Prokhorov et al. (2009). Let us note the reduced volume emissivity in the iron heliumlike emission lines as
where corresponds to the characteristic rate coefficient value (see also Prokhorov et al. 2009).
In Fig. 2 the reduced emissivity U, when the ionic fractions are in ionization equilibrium, is shown in the range of temperatures between 3.5 keV and 11 keV.
Figure 2: Dependence of the equilibriumreduced iron volume emissivity in the heliumlike lines on the plasma temperature. 

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In the situation considered above we find that at temperatures of kT_{1}=3.4 keV and kT_{2}=10.0 keV the equilibrium values of the reduced volume emissivities are approximately equal. However, in the presence of nonequilibrium processes, the approximate equality of these volume emissivities does not remain the same. Since the fraction of electrons with an energy higher than the impact excitation threshold keV is at the temperature kT_{1}=3.4 keV and is in turn much less than 72% at a temperature of kT_{2}=10.0 keV, the more effective impact excitation should be at a temperature of kT_{2}=10.0 keV. Furthermore, the nonequilibrium ionic fraction of heliumlike iron in the region of a temperature of kT_{2}=10.0 keV is higher than the equilibrium ionic fraction (see Fig. 1) and, therefore, nonequilibrium ionization leads to an increase of the volume emissivity in the heliumlike spectral lines.
Using the dependence of the ionic fractions of iron on the dimensionless time yr) (see Eq. (11)), we study the time evolution of the reduced volume emissivity in the iron heliumlike emission lines. This timeevolution is shown in Fig. 3.
Figure 3: Evolution of the reduced iron volume emissivity in the iron heliumlike lines in the region with a temperature of 10.0 keV. The dimensionless time is given by t/( yr). 

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Since the maximal value
of the
equilibriumreduced emissivity U(T) is at the temperature
keV and equal to 1.25 (see Fig. 2), we
conclude that the nonequilibrium value of U(t) (see Fig. 3) is higher than the maximal equilibrium value
up to a time
yr (up to a
dimensionless time equal to 0.65). The value
of
the equilibriumreduced emissivity at the temperature kT= 10.0 keV
equal to 0.8 is less than the nonequilibrium value of
U(t) during the time interval
yr. Therefore, the
iron abundance
A_{Z, eq}, derived from the assumption
that ionization states are in equilibrium, will be higher than the
correct iron abundance value A_{Z} (see Eq. (13)), which is given by
(14) 
4 An analysis of Helike and Hlike nonequilibrium ionization states by means of eigenvectors
In the previous section we showed that the ionic fraction of Helike
iron ions can remain in nonequilibrium while the Hlike iron ionic
fraction almost achieves equilibrium. This somewhat paradoxical
behavior can be more easily understood by the means of eigenvectors
of the transition matrix M, which is (see Eq. (11))
where yr is the age of the hydrodynamical structure (see Sect. 3).
Here we calculate the values of the eigenvalues of the transition matrix and the corresponding eigenvectors and show how the ionic fraction of Helike iron ions can remain in nonequilibrium longer than that of Hlike iron ions.
The eigenvalues
of the transition matrix M are derived
from the equation
(16) 
where E is the unit matrix.
One of the eigenvalues of the transition matrix M is of the form Eq. (15) equal zero ( ). Consequently, the ionization equilibrium is achieved in the end.
The solution of the system of differential equations (Eq. (11)) can be written as
where c_{ i} are constants, is the vector (n_{+23}, n_{+24}, n_{+25}, n_{+26}) and are the eigenvectors of the transition matrix M.
At the temperature kT=10.0 keV we derive three eigenvalues which
equal
,
and
.
Those eigenvectors which correspond to
the derived eigenvalues are respectively
(18) 
The eigenvector which corresponds to the eigenvalue determines equilibrium ionic fractions at a temperature of kT=10.0 keV.
Since is the smallest absolute value of the eigenvalues (excluding , which does not correspond to any relaxation process), the process which corresponds to the eigenvector is the slowest (see Eq. (17)). This slow process corresponds to the increase in the FeXXVII ionic fraction due to decreases in the FeXXV and FeXXVI ionic fractions. However, the absolute value of the second component of , which corresponds to the decrease in the FeXXV ionic fraction and equals 0.61, is higher than the absolute value of the third component of , which corresponds to the decrease in the FeXXVI ionic fraction and equals 0.14. Therefore, the variation in the heliumlike FeXXV ionic fraction which is proportional to the value of the second component of is more substantial during this relaxation process than the variation in the hydrogenlike FeXXVI ionic fraction.
Figure 4: Dependence of the Helike (solid line), Hlike (dashed line) and fully ionized (dotdashed line) ionic fractions of iron on the dimensionless time yr). 

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The quantitative difference between the variations in the Helike and Hlike ionic fractions is illustrated in Fig. 4.
5 Mach number determination
Clusters of galaxies form via gravitational infall and mergers of smaller mass concentrations. In the course of a merger, a significant portion of the kinetic energy of the colliding subclusters, which carried by the gas, is dissipated by shocks.
The shocks in the A520 and 1E065706 clusters have the Mach numbers
M=23, derived from the RankineHugoniot jumpconditions (for a
review, see Markevitch & Vikhlinin 2007), relating the density and
temperature jumps at the shock and the Mach number, M=v/c_{1},
where c_{1} is the velocity of sound in the preshocked gas and
v is the velocity of the preshock gas in the reference frame of
the shock. Thus, if the preshock and postshock temperatures
(T_{1} and T_{2}, respectively) are determined from
observations, the Mach number of the shock can be derived from the
equation:
(19) 
where is the adiabatic index. It is usually assumed that the preshock velocity in the reference frame of the shock is equal to the shock velocity in the reference frame of the galaxy cluster, and that the adiabatic index is (see Markevitch & Vikhlinin 2007).
The comparison of the Xray image and the gravitational lensing mass map of the 1E065706 merging cluster (Clowe 2006) shows that the mass peak of the subcluster is offset from the baryonic mass peak. Clowe et al. (2006) interpret this as the first direct evidence for the existence of dark matter.
Such merging clusters offer the unique opportunity to study gas physics through direct comparison of the observed shock properties with the predictions of gas and dark matter modeling (e.g. Prokhorov & Durret 2007; Springel & Farrar 2007). In this section we provide a new way to derive shock parameters based on measurements of the flux ratio of the FeXXV and FeXXVI iron lines.
The fluxes of the FeXXV and FeXXVI lines have the same dependence on the metal abundance as on the emission measure, their ratio is independent of these parameters. This iron line ratio can therefore be used to determine the temperature of the intracluster gas (e.g. Nevalainen et al. 2003) and the presence of suprathermal electrons (e.g. Prokhorov et al. 2009).
Taking into account both electronimpactexcitation and radiative
recombination, the iron line flux ratio is given by
(20) 
where the rate coefficients are , , and are the impactexcitation rates. The excited states b correspond to the upper levels of the Helike triplet and the Hlike doublet, and the radiative, branching ratios are given by,
(21) 
and and are the rate coefficients for the contribution from radiative recombination to the spectral lines FeXXV (Helike triplet) and FeXXVI (Hlike doublet), respectively, and are the transition probabilities.
Below we study the situation which was considered in Sects. 3 and 4. The variation of the iron line flux ratio in the region with temperature 10.0 keV as a function of dimensionless time is shown in Fig. 5.
Figure 5: Evolution of the iron line flux ratio in the region with a temperature of 10.0 keV. The dimensionless time is given by t/( yr). 

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Let the downstream velocity of the shocked gas which flows away from the shock be . Then the distance covered by the shocked gas is passed in a time t and is given by . Therefore, if the iron line flux ratio R is known from observations at a distance L from the shock front, we can derive the value of the downstream velocity using the function R(t), where t(R) is the inverse function for R(t).
The Mach number of the shock and the downstream velocity are
related by (e.g. Landau & Lifshitz 1959)
where and c_{2} is the velocity of sound in the postshocked gas.
On the observational side, it will be important to derive the flux
ratio of the FeXXV and FeXXVI iron lines from the region between the
shock front and the considered distance L which shocked gas covers
in a time
.
Since nonequilibrium ionization
can occur at a linear scale of 50 kpc (see Sect. 2), we
choose L=25 kpc. The flux ratio of the iron lines FeXXV and FeXXVI
from this region is then written as
using Eqs. (22) and (23), we find the Mach number Mof the shock as a function of the iron line flux ratio R. The dependence M(R) is plotted in Fig. 6.
Figure 6: Dependence of the Mach number of the shock M on the iron line flux ratio R. 

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Thus, we conclude that the Mach number of the shock can be derived from the iron line flux ratio. Methods based on the RankineHugoniot jump conditions and on measuring the iron line flux ratio are independent for deriving the Mach number of the shock.
6 Conclusions
The currently operating Xray imaging observatories provide us with a detailed view of the intracluster medium in galaxy clusters. Chandra's 1 angular resolution, the best among the current Xray observatories, corresponds to linear scales of <1 kpc at z<0.05 and 4 kpc at z=0.3 (the redshift of the 1E065756 cluster). This enables us to study hydrodynamic phenomena in galaxy clusters, such as shock waves and hot plasma bubbles.
Metal observations are always limited by the number of Xray photons. For diffuse low surface brightness objects like galaxy clusters, the effective area is a major issue, therefore in order to keep to a reasonable observation time, Chandra metal abundance maps will have lower spatial resolution than XMMNewton (see Werner et al. 2008). IXO^{} is planned to be a followup mission of XMMNewton and will have a sensitivity much higher than XMMNewton. The expected effective area of the IXO mirror and focal plane instruments showing the large improvement at all energy levels (including the 67 keV band) in comparison with those of current Xray observatories is plotted in a figure^{}. The larger effective area in the 67 keV band the higher the accuracy of the iron line flux measurements achieved. With IXO we will be able to resolve for the first time the metal distribution in the ICM on the scales of single galaxies in nearby clusters (simulations of metallicity maps, which will be provided by the next generation Xray telescope are given by Kapferer et al. 2006). Therefore, the sensitivity of IXO will provide metal observations near shock fronts and in hot plasma bubbles.
We have considered in this paper the nonequilibrium ionization at linear scales of 50 kpc in galaxy clusters. The necessary condition on the baryonic overdensity (see Eq. (8)) for the existence of nonequilibrium ionization in regions of galaxy clusters where holds for ions of iron. The reason for this is that the iron atomic number Z = 26 is high enough to reach the threshold value for the overdensity, which is proportional to Z^{3} (see Eq. (8)) and can therefore become higher than the mean cluster baryonic overdensity.
The dependence of the Helike and Hlike ionic fractions of iron on time is given in Sect. 3. We found that the Helike ionic fraction of iron does not achieve its equilibrium value during the age of the hydrodynamical structures, and nonequilibrium ionization takes place.
We calculated the reduced emissivity in the Helike iron spectral lines and concluded that the iron abundance derived from the assumption that ionization states are in equilibrium, is predicted to be higher than the correct iron abundance value (see Fig. 3).
We found that the slowest relaxation process corresponds to the increase in the FeXXVII ionic fraction due to decreases in the FeXXV and FeXXVI ionic fractions. However, the decrease in the FeXXV ionic fraction is much higher than the decrease in the FeXXVI ionic fraction during this relaxation process.
A new way to derive the Mach number of a shock based on measurements of the flux ratio of the FeXXV and FeXXVI iron lines is proposed in Sect. 5. The advantage of this method over the method based on the RankineHugoniot jump conditions is that ours is more accurate. Fortunately the iron line flux ratio is constrained without the effect of the hydrogen column density ( ) uncertainties. In practice, the Xray data can be fitted in a narrow band containing the FeXXV and FeXXVI lines, where the absorption is negligible (see Nevalainen et al. 2009). The drawback is that the number of photons is small in this narrow energy band, but the nextgeneration Xray telescope IXO with larger effective area overcomes this drawback and will be ableto measure the flux ratio of the iron K lines and, therefore, the Mach number of a shock with high precision. Using the narrow energy band instead of the full Xray spectrum minimizes the dependence on calibration accuracy (see Nevalainen et al. 2003), therefore the FeXXV to FeXXVI lines are insensitive to the details of the effective area function compared to the continuum spectrum. Note that the method based on the RankineHugoniot jump conditions uses the densities and temperatures derived from the continuum spectrum.
Another advantage of the proposed method is that it permits us to determine independently the Mach number of a shock by using measurements of the iron line flux ratio at different distances from a shock (see Sect. 5), since it takes into account an evolution of ionization states.
The effect of the apparent iron overabundance under the assumption of an ionization equilibrium and the slow process of heliumlike ionic fraction relaxation should be analyzed in galaxy clusters by means of future Xray observatories and may have implications in different astrophysical plasmas (e.g. in supernova remnants). New highspectralresolution instruments with higher sensitivity, such as IXO, are needed to measure the flux ratio of the iron Klines with the purpose of being independent in determining the shock parameters.
AcknowledgementsI am grateful to Joseph Silk, Florence Durret, Igor Chilingarian and Anthony Moraghan for valuable suggestions and discussions and thank the referee for very useful comments.
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Footnotes
 ...^{}
 The exact value of the ionized potential of a Helike ions is obtained by changing Z with . We consider the case and, therefore, the approximate value is sufficient.
 ... IXO^{}
 http://ixo.gsfc.nasa.gov/
 ... figure^{}
 http://ixo.gsfc.nasa.gov/images/science/effectivearea.jpg
All Figures
Figure 1: Dependence of the Helike (solid line) and Hlike (dashed line) ionic fractions of iron on the dimensionless time yr). 

Open with DEXTER  
In the text 
Figure 2: Dependence of the equilibriumreduced iron volume emissivity in the heliumlike lines on the plasma temperature. 

Open with DEXTER  
In the text 
Figure 3: Evolution of the reduced iron volume emissivity in the iron heliumlike lines in the region with a temperature of 10.0 keV. The dimensionless time is given by t/( yr). 

Open with DEXTER  
In the text 
Figure 4: Dependence of the Helike (solid line), Hlike (dashed line) and fully ionized (dotdashed line) ionic fractions of iron on the dimensionless time yr). 

Open with DEXTER  
In the text 
Figure 5: Evolution of the iron line flux ratio in the region with a temperature of 10.0 keV. The dimensionless time is given by t/( yr). 

Open with DEXTER  
In the text 
Figure 6: Dependence of the Mach number of the shock M on the iron line flux ratio R. 

Open with DEXTER  
In the text 
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