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Electromagnetic thermal instability with momentum and energy exchange between electrons and ions in galaxy clusters

Thermal instability in an electron-ion magnetized plasma, which is relevant in the intragalactic medium (IGM) of galaxy clusters, solar corona, and other two-component plasma objects is investigated. We apply the multicomponent plasma approach when the dynamics of all species is considered separately through the electric field perturbations. General expressions for the dynamical variables obtained in this paper can be applied for a wide range of astrophysical and laboratory plasmas also containing neutrals and dust grains. We assume that background temperatures of electrons and ions are different and include the energy exchange in the thermal equations for the electrons and ions along with the collisional momentum exchange in the equations of motion. We take into account the dependence of collision frequency on the density and temperature perturbations. The cooling-heating functions are taken for both electrons and ions. A condensation mode of thermal instability has been studied in the fast sound speed limit. A new dispersion relation including the different electron and ion cooling-heating functions and other effects mentioned above has been derived and its simple solutions for growth rates in the limiting cases have been found. We have shown that the perturbations have an electromagnetic nature. The crucial role of the electric field perturbation along the background magnetic field in the fast sound speed limit has been demonstrated. We have found that at conditions under consideration, the condensation must occur along the magnetic field while the transverse scale sizes can be both larger and smaller than the longitudinal ones. The results obtained can be useful for interpretation of observations of dense cold regions in astrophysical objects.

In majority of papers studying the thermal instability of astrophysical objects in the magnetic field, one uses the one-fluid ideal magnetohydrodynamics (MHD). In some papers, the two-fluid model with the ideal magnetic induction equation has been treated (e.g., Fukue & Kamaya 2007;Inoue & Inutsuka 2008). The non-ideal effects in the induction equation have been included in (e.g., Heyvaerts 1974;Stiele et al. 2006;Shadmehri et al. 2010).
In the plasma physics literature devoted to the thermal instability, one considers besides the one-fluid (e.g., Bora & Taiwar 1993) and two-fluid (e.g., Birk 2000) ideal MHD (from the point of view of the form of the magnetic induction equation) also a multicomponent media where the presence of electrons, ions, dust grains, and neutrals is taken into account (Kopp et al. 1997;Birk & Wiechen 2001;Pandey & Krishan 2001;Pandey et al. 2003;Shukla & Sandberg 2003;Kopp & Shchekinov 2007). Such effect as dust charge variation is also included when studying the thermal instability (Ibáñez & Shchekinov 2002;Pandey & Krishan 2001;Pandey et al. 2003). Analytical investigation of thermal instability in the multicomponent magnetized media with such physical effects as collisions between different species, ionization and recombination, dust charge dynamics, gravity, self-gravity, rotation, and so on is a very challenging problem. General basic equations have a very complex form (see, e.g., Kopp et al. 1997). Therefore, one usually treats simplified models considering, for example, potential perturbations in nonmagnetized (Kopp et al. 1997;Pandey & Krishan 2001;Ibáñez & Shchekinov 2002;Pandey et al. 2003;Shukla & Sandberg 2003;Kopp & Shchekinov 2007) and magnetized (Kopp et al. 1997;Shukla & Sandberg 2003) plasmas.
Neglecting the energy exchange between species in the thermal equations is one of the usual simplifying assumptions. It is justified when a collisional coupling is weak or when the energy exchange frequency is sufficiently large, so that the temperatures of species are equal. In a general case, one can obtain considerable complications. Nevertheless, some authors took into account the energy exchange for thermal instability in the two-fluid MHD framework (e.g., Birk 2000;Birk & Wiechen 2001).
The terms describing the energy exchange contribute to the dispersion relation not only through the temperature (or pressure) perturbations (Birk 2000;Birk & Wiechen 2001) but also through the perturbation of collision frequency which depends on the density and temperature. This effect is important when background temperatures of species are different. Such a situation can occur, for example, in galaxy clusters (see Markevitch et al. 1996;Fox & Loeb 1997;Ettori & Fabian 1998;Takizawa 1998).
In the present paper, we investigate the thermal instability in the electron-ion magnetized plasma which is relevant in the IGM of galaxy clusters, solar corona, and other two-component plasma objects. We apply the multicomponent plasma approach when the dynamics of all species is considered separately through the electric field perturbations (the E-approach; see, e.g., Nekrasov b, 2009Nekrasov & Shadmehri 2010, 2011.
General expressions obtained in this paper can be applied for a wide range of astrophysical and laboratory plasmas. We assume that the background electron and ion temperatures are different and include the energy exchange in the thermal equations for electrons and ions.
We take into account the dependence of collision frequency on the density and temperature perturbations. Different cooling-heating functions are assumed for electrons and ions. We include neither ionization and recombination effects nor gravity. Some expressions for electron and ion perturbations are obtained in the general form which can be used for other species (dust grains and neutrals). Here, we treat the condensation mode of thermal instability in the fast sound speed limit. We derive the general dispersion relation taking into account the effects mentioned above and find its simple solutions for the growth rates in the limiting cases.
The paper is organized in the following manner. In section 2, we give the fundamental equations used in this paper. The equilibrium state is considered in section 3. Equations for temperature perturbations are obtained in section 4. In section 5, we give specific conditions for further consideration. In section 6, equations for components of velocity perturbations are given in the fast sound limit. Components of the perturbed current are calculated in section 7. These components for the simplified collision contribution are given in section 8.
In section 9, we derive the dispersion relation. Its limiting cases are considered in section 10. We discuss the obtained results in section 11. The possible astrophysical implications are considered in section 12. Summing up of the main points is given in section 13.

BASIC EQUATIONS
The fundamental equations that we consider here are the following: the equation of motion, the continuity equation, and ∂T e ∂t +v e ·∇T e +(γ − 1) T e ∇·v e = − (γ − 1) 1 n e ∇ · q e −(γ − 1) 1 n e L e (n e , T e )−ν ε ei (n i , T e ) (T e − T i ) are the temperature equations for ions and electrons. In Equations (1) and (2), the index j = i, e denotes the ions and electrons, respectively. The force F j in Equation (1) is given by Other notations in Equations (1)-(5) are the following: q j and m j are the charge and mass of species j = i, e, v j is the hydrodynamic velocity, n j is the number density, p j = n j T j is the thermal pressure, T j is the temperature, ν ie (ν ei ) is the collision frequency of ions (electrons) with electrons (ions), ν ε ie (n e , T e ) = 2ν ie (ν ε ei (n i , T e )) is the frequency of the thermal energy exchange between ions (electrons) and electrons (ions) (Braginskii 1965), n i ν ε ie (n e , T e ) = n e ν ε ei (n i , T e ), γ is the ratio of specific heats, E and B are the electric and magnetic fields, and c is the speed of light in vacuum. The value q e in Equation (4) is the electron heat flux (Braginskii 1965). As for the latter, we will consider a weakly collisional plasma when the electron Larmor radius is much smaller than the electron collisional mean free path. In this case, the electron thermal flux is mainly directed along the magnetic field, where χ e is the electron thermal conductivity coefficient and b = B/B is the unit vector along the magnetic field. In other respects, a relation between cyclotron and collision frequencies of species stays arbitrary in general expressions considered below. We only take into account the electron thermal flux (6) because the corresponding ion thermal conductivity is considerably smaller (Braginskii 1965). We also assume that the thermal flux in the equilibrium is absent. The cooling and heating of plasma species in Equations (3) and (4) are described by function L j (n j , T j ) = n 2 j Λ j (T j ) − n j Γ j , where Λ j and Γ j are the cooling and heating functions, respectively. The form of this function has some difference from the usually used cooling-heating function £ (Field 1965). Both functions are connected with each other via equality L j (n j , T j ) = m j n j £ j . Our choice is analogous to that as in (Begelman & Zweibel 1994;Pandey & Krishan 2001;Shukla & Sandberg 2003;Bogdanović et al. 2009;Parrish et al. 2009). The function Λ j (T j ) can be found, for example, in (Tozzi & Norman 2001).
Electromagnetic equations are Faraday's and Ampere's laws, where j = j q j n j v j . We consider wave processes with typical timescales much larger than the time the light spends to cover the wavelength of perturbations. In this case, one can neglect the displacement current in Equation (8) that results in quasi-neutrality for both the electromagnetic and purely electrostatic perturbations. The magnetic field B includes the background magnetic field B 0 , the magnetic field B 0cur of the background electric current (when it presents), and the perturbed magnetic field.
For generality, we assume in the meanwhile that n i and n e are different, having in mind that some solutions obtained below can be applied for multicomponent plasmas.

EQUILIBRIUM STATE
At first, we will consider an equilibrium state. We assume that the background velocities of species are absent. Here, we do not involve an equilibrium inhomogeneity.
Then, the thermal equations (3) and (4) in the equilibrium take the form

LINEAR EQUATIONS FOR THE TEMPERATURE PERTURBATIONS
We now consider Equations (3) and (4) in the linear approximations. Using Equations (2) and (9) for ions, we find where the index 1 denotes the perturbed values. The following notations are here introduced: where the index 0 denotes the unperturbed values. Analogously, we obtain for the electrons where In notations (11) and (13), we have introduced the following frequencies: , We assume that the background magnetic field B 0 is directed along the z-axis. In notations (11) and (13), we have used an equilibrium state and the fact that ν ε ei (n i0 , T e0 ) ∼ n i0 and ν ε ie (n e0 , T e0 ) ∼ n e0 . We see from Equations (10) and (12) that the temperature perturbations are connected with a velocity divergence. Solutions for T e1 and T i1 are given by where the following notations are introduced: To find the temperature perturbation T j1 , we have to calculate expressions for ∇ · v j1 .
General equations for the velocity v j1 and ∇ · v j1 are derived in the Appendix, where expressions for D and G l , l = 1, 2, 3, 4, are also given. In their general form, the components of v j1 are very complex. Therefore to proceed further analytically, here we restrict ourselves to a limiting case in which the dynamical frequency ∂/∂t is smaller than the sound frequency. Thus, we will consider sufficiently short-wavelength perturbations along the magnetic field (see below). Some additional simplifying conditions which are satisfied in magnetized plasmas are also used.

SPECIFIC CASE: FAST SOUND SPEED LIMIT
Equations (A27), (A28) and (A30) are written in their general form which allows us to consider different simplified specific cases corresponding to the real astrophysical conditions.
We further consider the case in which Since usually ω 2 ci ≫ ∂ 2 /∂t 2 in magnetized plasmas, condition (18) is satisfied for a wide range of the transverse wavelengths of perturbations (∂/∂y) −1 in comparison with the longitudinal wavelengths (∂/∂z) −1 . The other simplifying condition is This inequality means that the effective ion Larmor radius is much smaller than the perturbations wavelengths.
Below, we consider the case in which the perturbation frequency is much smaller than the sound frequency, i.e., This condition can be written in the form ∂ 2 /∂t 2 ≪ c 2 s ∂ 2 /∂z 2 , where c s can be considered as the effective sound velocity. Inequalities (19) and (20) are given in the approximate form to unite two cases, ∂ ∂t > (<)Ω ie,ei . We note that the case (18) is followed from conditions (19) and (20).
Under conditions (19) and (20), we find, using notations (A10), (A30), and (A31), the following equations for P i,e1 (see Equations (A27) and (A28)): where Below, we will need the correction terms proportional to α e,i ≪ 1 and β e,i ≪ 1. We note that the contribution of the correction term proportional to Q e1y in Equation (21) is at least of the order of m e /m i in comparison with the term proportional to Q i1y . However for convenience, we keep this small term for the symmetry with equation for P e1 . The ratio of contributions of the correction terms proportional to F e1z and F i1z in Equation (21) is of The same for terms proportional to Q i1y and Q e1y in Equation (22) is and for the ratio of terms ∼ F i1z and F e1z , we have We now obtain equations for components of velocities v i,e1 , using Equations (21) and (22).

Equations for v i,e1y
From Equations (A3), (21), and (22), we find, using notations (A6), We see that these equations are obtained one from another by changing i ↔ e. We note that α j , β j ∼ m j . However, we keep in these equations some small terms for the symmetry of both equations. The terms proportional to ω −4 cj are needed in equations for v i,e1x . We also note that For these estimations, we have used expressions (A18), (A25), (A26), and (A29), assuming W j ∼ ∂/∂t.

Equations for v i,e1x
Equations for v i,e1x are found from Equation (A2) by using Equations (24) and (25).

Equations for v i,e1z
From Equations (A7), (21), and (22), we obtain, using notations (A6), We note that in Equation (26) ( (27)) the terms proportional to β e F e1x,y (α e F e1x,y ) are small as compared with α i F i1x,y (β i F i1x,y ). However, we also keep them for the symmetry of these equations.

COMPONENTS OF CURRENT
We now find components of the linear current j 1 = j q j n j0 v j1 . It is convenient to calculate the value 4π (∂/∂t) −1 j 1 . We further consider the electron-ion plasma in which n e0 = n i0 , q e = −q i . In our calculations, we will use an equality m e ν ei = m i ν ie . From Equations (A2), (24)- (27), and (5) in the linear approximation, we find Here, the following notations are introduced: where ω pi = (4πn i0 q 2 i /m i ) 1/2 is the ion plasma frequency. The value δ in expression for a zz has the form

SIMPLIFICATION OF COLLISION CONTRIBUTION
The relationship between ω ce and ν ei or ω ci and ν ie (that is the same) can be arbitrary in Equations (28)-(30) (except of that in the thermal conduction). We further proceed by taking into account that ∂/∂t ≪ ω ci . In this case, we can neglect the collisional terms proportional to b xy and b yx (see notations (31)). However, a system of Equations (28)-(30) remains sufficiently complex to find j 1 through E 1 . Therefore, we further consider the case in which the following condition is satisfied: It is clear that condition (33) can easily be realized. In this case, we may neglect collisional terms proportional to b x,zx and b y,zy . Then, the system of Equations (28) Here, Parameter d defines the collisionless, d ≪ 1, and collisional, d ≫ 1, regimes. Below, we will derive the dispersion relation.

DISPERSION RELATION AND ELECTRIC FIELD POLARIZATION
We will further consider Equations (34)-(36) in the Fourier-representation, assuming that perturbations have the form exp (ikr−iωt). Then using Equations (7) and (8), we obtain the following system of equations: where E 1k is the Fourier-image of the electric field perturbation, n = kc/ω. The index k by E 1k is equal to k = {k,ω}. For the Fourier-images of operators ε ij and d, we keep the same notations. In general, we see that the longitudinal electric field E 1zk ∼ E 1x,yk inevitably arises when k y = 0 and n 2 y − ε zz = 0. The dispersion relation can be found by setting the determinant of the system (39) equal to zero. Contribution of terms proportional to ε xy and ε yx into dispersion relation is small as compared with that of terms ∼ ε xx , ε yy according to condition (18) and α i , β i ≪ 1. Neglecting these small terms, we obtain n 2 z − ε yy E 1yk + (−n y n z + ε yz ) E 1zk = 0, ε zx E 1xk + (−n y n z + ε zy ) E 1yk + n 2 y − ε zz E 1zk = 0.
Below, we will consider the dispersion relation for this system in the collisionless (d ≪ 1) and collisional (d ≫ 1) cases.

Collisionless case
Consider the case where an estimation of d is d ∼ ν ie ω/k 2 z c 2 s for δ ∼ α i /m i . Then the dispersion relation has two solutions when n 2 z − a yy = 0 and n 2 z − a yy = 0 (see the second equation in the system (40)).
The case n 2 z − a yy = 0 This case describes the Alfvén waves ω 2 = k 2 z c 2 A , where c A = B 0 / (4πn i m i ) 1/2 . Polarization of the electric field has the form E 1yk = 0 and The case n 2 z − a yy = 0 In this case, the dispersion relation reduces to ε zz ≃ 0. Contribution of all other terms can be shown, using expressions (31) and (37), to be small.

The last equation means that
Polarization of these perturbations is the following: We see that the electric field in the plane of the wave vector is a potential one. However, the component E 1xk can be large, E 1xk ∼ α i (k y k z ω ci /k 2 ω) E 1zk . Thus, in general, perturbations have an electromagnetic nature.

Collisional case
In the collisional case, the dispersion relation of the system (40) takes the form n 2 z − a yy a zz = 0.
We see from Equation (45) that the same cases, n 2 z − a yy = 0 and a zz = 0, are possible analogously to the case d ≪ 1 considered above. However, polarization in the case n 2 z − a yy = 0 is different: E 1zk = 0, E 1yk = 0, and E 1xk = (n y n z d/a zx ) E 1yk ∼ (ν ie /ω ci ) E 1yk . This is a mixture of the Alfvén and magnetosonic waves. For the dispersion relation a zz = 0, the electric field polarization is the following: where an estimation for d is given above. Thus, the electric field perturbation has all three components.
We see that in the case ε zz = 0 the electric field perturbation along the magnetic field plays the crucial role in the fast sound speed regime (20).

SOLUTION OF DISPERSION RELATION ε zz = 0
We now consider the dispersion relation (42) which is suitable also for the collisional case. Using notations (23), (32), and (A16)-(A20), we obtain the following dispersion relation: where ∂/∂t = −iω and Ω χ = (γ − 1) (χ e0 /n e0 ) k 2 z . All terms Ω are defined by the system (14). We see that the first four terms on the left hand-side of Equation (47) are symmetric concerning the contribution of electrons and ions. The third and fourth terms have appeared due to dependence of collision frequency ν ε ei,ie (n i,e0 , T e ) on density perturbation. The last three terms are connected with perturbation of this frequency because of the electron temperature perturbation. All terms proportional to Ω ei,ie and Ω T ei,ie are connected with the energy exchange in the thermal equations (3) and (4). We see that taking into account the collision frequency perturbation in a general case T e0 = T i0 results in considerable modification of the dispersion relation. However, this effect should be involved because the absence of the thermal equilibrium between electrons and ions can occur, for example, in galaxy clusters (e.g., Markevitch et al. 1996;Fox & Loeb 1997;Ettori & Fabian 1998;Takizawa 1998). We further use the fact that n i0 = n e0 in our case and ν ε ei,ie ∼ T −3/2 e (Braginskii 1965). Then, we obtain Ω ei = Ω ie and Ω T ei = Ω T ie = −3Ω ie (T e0 − T i0 ) /2T e0 . In this case, Equation (47) takes the form Below, we will consider different limiting cases of Equation (48).
10.1. The case Ω ie = 0 If we do not take into account the energy exchange, Ω ie = 0, and set T e0 = T i0 , then we obtain equation It is followed from Equation (49) that the ion cooling-heating function modifies the growth rate which takes place without this function. Neglecting the contribution of the ion cooling and heating, Ω T i = Ω ni = 0, we have It is easy to see that this solution is a mixture of isochoric and isobaric solutions (Parker 1953;Field 1965) because we have taken into account the ion temperature perturbation. If we neglect the latter, i.e. neglect the second term ∼ T i0 in Equation (48), we obtain the usual isobaric solution We also see from Equation (48) that for the short-wavelength perturbations when Ω χ ≫ ω, Ω T e , Ω ne the thermal instability can arise due to the ion cooling function

The case Ω ie = ∞
When the frequency Ω ie is much larger than other frequencies, 2Ω ie ≫ ∂/∂t, Ω χ , Ω T e,i , and T e0 = T i0 , then the dispersion relation becomes the following: This is isobaric solution with the electron and ion cooling.
For the different temperatures of electrons and ions, T e0 = T i0 , we obtain In the case T e0 ≫ T i0 , this equation takes the form, In the opposite case, T e0 ≪ T i0 , we obtain

General case
In a general case, Equation (48) can be written in the form where Equation (52) can be solved numerically for different cooling functions.

DISCUSSION
From the results obtained above, we can estimate the relative perturbations of the number density and pressure in the fast sound speed regime. Using equations (24)-(27) and keeping the main terms, we find expressions for ∇ · v j1 , Thus, the number density perturbation n j1 ∼ ∇ · v j1 is determined by the longitudinal electric field E z . If we use the condition of quasineutrality, ∇ · v i1 = ∇ · v e1 , and relation where the value P j1 is connected with the pressure perturbation through the relation From the linear continuity equation and Equations (54)-(56), we find It is followed from here the relation between p i1 and p e1 , The sum of pressures, p e1 + p i1 , is equal to Thus, the total pressure almost does not change. An estimation of the value α i /m i − β e /m e gives (see notations (23)) . Thus, we obtain n i1 /n 0 ∼ p i1 /p 0 .
Our dispersion relation (47)  which does not contain the magnetic field. Therefore, a wide spectrum of wavelengths along and across the magnetic field can be generated in the framework of conditions (18)- (20) and (33). It is worth to note that condition (20) is a particular case which is considered in this paper. Other conditions used above are easily satisfied in many astrophysical objects -25because the ion cyclotron frequency is much larger than the dynamical and ion-electron collisional frequencies and the ion Larmor radius is much smaller than the perturbations wavelengths.
The general form of the dispersion relation (48) including the thermal exchange between electrons and ions and their different cooling-heating functions and temperatures permits us to consider various cases which can be realized in real situations. In particular, Equation (49) is available for a weak thermal coupling and equal temperatures, while Equations (50) and (51) are appropriate in the case of strong coupling. The intermediate case is described by Equation (52), where coefficients (53) contain both different temperatures and cooling functions. The large thermal conductivity Ω χ stabilizes some perturbations.
We note that one particular Field length (Field 1965) can be obtained from the condition Ω χ = −Ω T e + Ω ne . However, in the limit Ω χ ≫ ω, Ω T e,i , Ω ne,i , 2Ω ie , Equation (48) describes instability due to the ion cooling (see section 10).
We have shown that unstable perturbations have an electromagnetic nature (see Equations (43) and (46)). Thus, a consideration of potential perturbations given in some papers (see section 1) is not adequate. (d ≫ 1), the dispersion relation has the same form ε zz = 0.

ASTROPHYSICAL IMPLICATIONS
The fundamental purpose of this paper is to investigate the thermal instability taking into account the real multicomponent nature of medium appropriate, e.g., for galaxy clusters in a straightforward manner, using the E-approach. Therefore, the different cooling-heating functions for the electrons and ions, the energy exchange between them, and perturbation of the energy exchange collision frequency for the non-equilibrium background state need to be appropriately included. The last effect considered for the first time for thermal instabilities contribute to the dispersion relation with the same order of magnitude as other effects and considerably modifies it (see, e.g., Equations (50) and (51)). We note that the effects mentioned above are not considered in the astrophysical literature using the MHD-approach. The key condition for the results obtained is condition (20). It denotes that we consider sufficienlty short-wavelengths along the magnetic field. In the simplest case without the energy exchange and ion temperature perturbations, the growth rate is in the region of the isobaric solution ω = − i γ (Ω χ + Ω T e − Ω ne ) (see section 10.1). The same growth rate for the condensation mode, we have, for example, for the neutral and magnetized media when the perturbation wavelength tends to infinity (e.g., Field 1965;Heyvaerts 1974;Loewenstein 1990;Balbus 1991 We see from the first condition that the relation between k ⊥ and k z can be arbitrary because ω 2 ci ≫ ω 2 , ν ie ω, i.e. the cases k ⊥ k z and k ⊥ ≫ k z are permitted. We emphasize that this conclusion is a consequence of the fast sound speed limit (20). The case k z → 0 is not suitable for this limit. The second condition means that the ion sound (or Larmor for Ω ie = 0) radius is much smaller than the transverse wavelength. For parameters appropriate for galaxy clusters, B 0 ∼ 1 µG and T ∼ 3 keV, we have ω ci ∼ 9.6 × 10 −3 s −1 and c s ∼ (2T /m i ) 1/2 ∼ 7.6 × 10 2 km s −1 (for protons). Thus, the lower limit to the transverse wavelength λ ⊥ = 2π/k ⊥ is equal to λ ⊥ > 5 × 10 5 km. We note that since the growth rates do not depend on k ⊥ there is no a specific connection between k ⊥ and k z for the most unstable modes. The upper limit to k z is determind by the thermal conductivity.
The dispersion relation (47) for condensation modes does not contain the magnetic field because the main dynamics of species in the case under consideration is along the magnetic field lines and determined by the longitudinal electric field E z . This field is not captured by the ideal magnetic induction equation. We have obtained the dispersion relation which cannot followed from the MHD consideration. At the same time, the simplest condensation mode growth rate described by Equation (47) coincides with the MHD result (see above this section).
We further shortly outline some important points of our investigation for possible observations. We have found the growth rates which contain the separate contribution of the cooling functions of electrons and ions in a clear form. It is obvious that both components (in multicomponent media, the dust grains and neutrals can also be present) can result in the thermal instability in the similar manner. This fact considerably extends possibilities for a medium to become unstable. In this connection, it is important to know the functional dependence of the cooling functions on the temperature and density for each species. The range of the scale lengths of unstable perturbations can enlarge due to contribution of other species except electrons to the instability. For example, the short-wavelength perturbations which must be stable because of the large electron thermal conduction can be unstable due to contribution of ions to cooling of medium (see section 10). In the fast sound regime when the growth rates do not depend on the magnetic field and wavelengths of perturbations, both filaments and clouds and pancakes can be observed. It is important to have in mind that in this regime the electric field and electric currents of species along the magnetic field lines have the crucial role. This result could be used for the diagnostics of the magnetic field direction in unstable domains.

CONCLUSION
We have treated the thermal instability in the electron-ion magnetized plasma which is relevant to galaxy clusters, solar corona, and other two-component plasma objects. The multicomponent plasma approach have been applied to derive the dispersion relation for the condensation modes in the case in which the dynamical frequency is much slower than the sound frequency. Our dispersion relation takes into account the electron and ion cooling-heating functions, collisions in the momentum equations, energy exchange in the thermal equations, different background temperatures of the electrons and ions, and perturbation of the energy exchange frequency due to density and temperature perturbations. Different limiting cases of the dispersion relation have been considered and simple expressions for the growth rates have been obtained. We have shown that perturbations have an electromagnetic nature. The important role of the electric field perturbation along the background magnetic field has been demonstrated. We have found that at conditions under consideration, the condensation must occur along the magnetic field lines while the transverse scale sizes can be both larger and smaller than the longitudinal ones. General expressions for the dynamical variables obtained in this paper can be applied for a wide range of astrophysical and laboratory plasmas also containing the neutrals and dust grains. The results obtained can be useful for interpretation of observations of the dense cold regions in astrophysical objects such as IGM, solar corona, and so on.
I would like gratefully to acknowledge Dr. Mohsen Shadmehri for valuable discussions and suggestions and also the anonymous referee, whose useful comments helped in improving the manuscript.

A.1. Perturbed velocities of species
In the linear approximation, Equation (1) for the perturbed velocity v j1 takes the form where p j1 = n j0 T j1 + n j1 T j0 . From this equation, we can find solutions for the components of v j1 . For simplicity, we assume that ∂/∂x = 0 because a system is symmetric in the transverse direction relative to the z-axis. Then, the x-component of Equation (A1) gives where ω cj = q j B 0 /m j c is the cyclotron frequency of species j. Differentiating Equation (A1) over t and using Equation (2) in the linear approximation and Equations (15), (16), and (A2), we obtain for the y-component of Equation (A1) where The value P j1 is connected with the pressure perturbation (see Equation (A1)). Using Equations (A2) and (A3), we find In Equations (A3) and (A5), notations Q j1x = ω cj F j1y + ∂F j1x ∂t are introduced. We see from these equations that the thermal pressure effect on the velocity v i1x is much larger than that on v i1y when ∂/∂t ≪ ω ci . The z-component of Equation (A1) can be written in the form ∂ 2 v j1z ∂t 2 = ∂P j1 ∂z + ∂F j1z ∂t . (A7) A.2. Calculation of ∇ · v j1 and P j1 We have Using Equations (A3), (A4), (A7) and (A8), we obtain Here, and operators L 1j and L 2j are the following: L 3j = ∂ 4 ∂y 2 ∂t 2 + ∂ 2 ∂t 2 + ω 2 cj ∂ 2 ∂z 2 .
From a system of equations (A9), we find The values P i1 and P e1 can be found, substituting solutions (A12) into expressions (A4), Using expressions (A16), (A17), (A20), and (A24), we obtain for the operator K (A23) the simple form, where notations are introduced. Using Equations (A21) and (A22), Equation (A14) for P i1 takes the form Analogous consideration of Equation (A15) leads to the following equation for P e1 :