Dissipative models of colliding stellar winds I. Effects of thermal conduction in wide binary systems

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1 Mon. Not. R. Astron. Soc. 300, (1998) Dissipative models of colliding stellar winds I. Effects of thermal conduction in wide binary systems A. V. Myasnikov 1 * and S. A. Zhekov 2 * 1 Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadskii Ave., Moscow , Russia 2 Space Research Institute, Bulgarian Academy of Sciences, 6 Moskovska str., Sofia 1000, Bulgaria Accepted 1998 May 28. Received 1998 May 27; in original form 1998 January 5 1 INTRODUCTION It is well known that early-type stars (O, A, B), Wolf Rayet (WR) stars, and pre-main-sequence (PMS) stars (T Tauri, Herbig Ae/Be etc.) undergo mass loss (stellar winds, jets, outflows), and that the interaction of the stellar winds with the circumstellar environment is one of the many manifestations of this activity. Additionally, the percentage of binary and multiple systems among the stellar types listed above is relatively high ( 40 per cent for the WR stars, Conti 1988; > 67 per cent for the young stars, Leinert et al. 1993). These two properties binarity and strong winds suggest that studying colliding stellar winds (CSW) will be of great interest both from the theoretical and observational view points. The general physical picture of the CSW is as follows. There are three surface discontinuities: two shocks, each compressing the stellar wind of the corresponding binary component, and a contact discontinuity, separating the two compressed gas flows. Since the wind velocities are relatively high (e.g., for WR and O stars: V W km s ¹1, Prinja, Barlow & Howarth 1990), both parts of the interaction region will be filled by a high temperature plasma (> K). This high temperature suggests the observational manifestation of CSW, namely, they are expected to be powerful emitters of soft X-rays (0.2 4 kev). In fact, WR (WRþO) binaries have X-rays luminosities which are on average higher than those of single WRs (Pollock 1987). This extra source of X-ray emission is probably a result of the presence of CSW. Also, in the framework of the CSW model, an excess in the soft (E < 1 kev) part of the X-ray emission of the Herbig Ae/Be * myas@ipmnet.ru (AVM); szhekov@bgearn.acad.bg (SAZ) ABSTRACT The influence of electron thermal conduction on the 2D gas dynamics of colliding stellar winds is investigated. It is shown that, as a result of the non-linear dependence of the electron thermal flux on the temperature, the pre-heating zones (in which the hot gas in the interaction region heats the cool winds in front of the shocks) have finite sizes. The dependence of the problem of the structure of the flow in the interaction region on the dimensionless parameters is studied, and a simple expression is derived for the size of the pre-heating zones at the axis of symmetry. It is shown that small values of the thermal conductivity do not suppress the Kelvin Helmholtz instability if the adiabatic flow is subject to it. Further studies, both numerical and analytical, in this direction will be of great interest. The influence of thermal conduction on the X-ray emission from the interaction region is also estimated. Key words: hydrodynamics instabilities shock waves binaries: general stars: mass-loss. binaries was predicted (Zhekov, Palla & Myasnikov 1994), and it is intersting to mention that such an excess was detected (Preibisch & Zinnecker 1994). Moreover, some of the observed Herbig Ae/Be stars among those showing a soft X-ray excess are known binary systems. In recent years, not only the observations but also the theory of the CSW have made considerable progress by making use of different numerical techniques (Lebedev & Myasnikov 1988, 1990; Luo, McCray & Mac Low 1990; Myasnikov & Zhekov 1991,1993; Kallrath 1991; Stevens, Blondin & Pollock 1992, and others). However, the CSW model is not yet fully consistent with the observations and its physics is not yet complete. For example, the theoretical X-ray luminosities of the WRþO binaries, deduced from hydrodynamic modelling, are larger by 1 2 orders of magnitude than the observed values ( Zhekov & Myasnikov 1989; Stevens et al. 1992; Myasnikov & Zhekov 1993). Although Willis, Schild & Stevens (1995) and Stevens et al. (1996) have reported compatibility between the model and the observations, we would recall the difficulties encountered when modelling the radiative interaction flows with a stagnation point (see Myasnikov, Zhekov & Belov 1998). Also, the co-existence of high (T > 10 6 K, shock region) and low (T 10 4 K, stellar winds) temperature gases suggests that thermal conduction can play an important role in the physics of this system. This is because the temperature gradient is what determines the energy transfer in this case. It has to be mentioned that the electron thermal conduction is the most important process in the fully ionized plasma because of the much smaller mass of the electrons with respect to various ions. However, there is no numerical model that considers thermal conduction effects in CSW. In the present paper we start studying the influence of thermal 1998 RAS

2 Dissipative models of colliding stellar winds I 687 conduction on the CSW. We would like to mention that the effects of thermal conduction are expected to be more pronounced in the case of wide binaries because of the smaller gas density in the winds. This is the case because the typical conduction scalelength is proportional to the electron temperature equilibrium time, which in turn is inversely proportional to the gas number density. On the other hand, Myasnikov, Zhekov & Belov (1998) have recently pointed out some difficulties related to the highly radiative case of flows possessing a stagnation point, and the radiative CSW are exactly such a case. These authors suggested that a possible way to solve the problem with the singularity at the stagnation point is to consider a real dissipative process in the model of CSW. However, one should keep in mind that the presence of a magnetic field will greatly reduce the effect of thermal conduction in a direction perpendicular to the field lines. The interaction between the magnetic field and thermal conduction results in a quite complicated geometry. All this makes us confident that it is worthwhile to study the pure influence of the thermal conduction on the physics of CSW, in order to better distinguish its effects from others which might appear when a more complex model is constructed. Section 2 presents the hydrodynamical model, Section 3 refers to the numerical methods in use and the results are given in Section 4. Our conclusions are given in Section 5. 2 GAS DYNAMICAL MODEL Let us consider a CSW binary whose components have mass loss rates Ṁ 1 and Ṁ 2, terminal velocities V 1 and V 2, and the separation between the components is 2d. If we normalize all the linear sizes to d, all the velocities to V 1, and the densities and pressures to Ṁ 1 =4 V 1 d 2 and Ṁ 1 V 1 =4 d 2, respectively, a dissipative flow of fully ionized quasi-neutral plasma in the one-fluid and one-temperature approach with electron thermal conduction can be described by the dimensionless system of differential equations: r þ rv t ð Þ ¼ 0 rv þ rv vþpi ¼ 0 ð1þ t e t þ eþp v ¼¹Gthc q e where r, p, v and e are the gas density, pressure, velocity and total energy per unit volume, respectively; is the tensor product, I is the identity matrix, q e is the isotropic electron thermal conduction flux given by: q e ¼¹T 5=2 T(Spitzer 1962), T ¼ p=r is the dimensionless temperature. The dimensionless thermal conductivity can be written as m G thc ¼ 0:386 m ¹ m m3:5 V1ð8Þd 5 ð12þ Ṁ 1ð6Þ; ¹1 ð2þ where m is the mean atomic weight per particle, m is the mean atomic weight per nucleon, d ð12þ ¼ d=10 12 cm, V 1ð8Þ ¼ V 1 =10 8 cm s ¹1, Ṁ 1ð6Þ ¼ Ṁ 1 =10 ¹6 M yr ¹1. The gas dynamics of the interacting supersonic radial flows without thermal conduction has been studied in detail by Lebedev & Myasnikov (1988, 1990). In this case the numerical problem has a 2D axially symmetric geometry and the interacting supersonic sources are separated by the contact discontinuity S, and the interaction region is bounded by the shocks F and G (Fig.1). Regions III and IV, bounded by the surfaces H 1 and F, and the surfaces H 2 and G, respectively, consist of the pre-shocked gas flow which can be described with the help of a spherically symmetric Figure 1. A schematic diagram of the interaction region between thermally conducting CSW. F, G shock waves, S contact discontinuity, s sonic lines, f streamlines, P F and P G the boundaries of pre-heating zones, R 1;2 radii of the stars, 2d separation between stars. solution of the system ð1þ with q e ¼ 0: rur 2 1 ¼ 1 p ¼ 1 g rg K 2ðg¹1Þ g¹1 ðgþ1þ=2 1 gþ1 U 2 þ 2g p g¹1r ¼ 1 for region III and rur 2 2 ¼ Lx ¹1 p ¼ 1 g rg L 1¹g x 2g K 2ðg¹1Þ g¹1 2 gþ1 ðgþ1þ=2 U 2 þ 2g p g¹1r ¼ x2 for region IV. Here K 1;2 ¼ R 1;2 =d are dimensionless critical radii, x ¼ V 2 =V 1 is the ratio of the terminal wind velocities, L ¼ Ṁ 2 V 2 =Ṁ 1 V 1 is the ratio of the gas dynamical pressures, and g is the adiabatic index. If thermal conduction comes into play, a pre-heating zone appears in front of the shock where the upstream electrons are being heated before crossing the shock. Following Zel dovich & Raizer (1968), we can estimate the size of the pre-heating zones at ð3þ ð4þ

3 688 A. V. Myasnikov and S. A. Zhekov the axis of symmetry under the assumption that the flow in front of the shock can be considered as an incompressible 1D steady-state flow (the latter is equivalent to assuming that the thermal conduction is the main energy transport mechanism, so we can also apply the resultant formulae to time-dependent flows such as wind-blown bubbles). In this case the third equation of the system ð1þ can be rewritten as ru g ¹ 1 ¼ G thct 3 dt 2 dy ; where T is the dimensionless temperature and y is the dimensionless distance from a star to the corresponding shock. Substituting the value ru from equation ð3þ, one can obtain for the pre-heating zone that 1 g ¹ 1 y y¹z F dy y 2 ¼ G thc Ts 0 T 3 2 dt; where Z F is the dimensionless distance from the shock to the point at which the flow temperature is equal to zero, that is, Z F is the size of the pre-heating zone; T s is the gas temperature at the shock front. Integrating this equation under the assumptions that g ¼ 5=3 and T s ¼ 2ðg ¹ 1Þ=ðg þ 1Þ 2 is equal to the adiabatic shock temperature, we obtain Z F ¼ 4:06 10¹3 F0G 2 thc 1 þ 4:06 10 ¹3 ð5þ F 0 G thc for the pre-heating zone between regions I and III, and similarly Z G ¼ 4:06 10¹3 ð2 ¹ G 0 Þ 2 G thc x 6 =L 1 þ 4:06 10 ¹3 ð2 ¹ G 0 ÞG thc x 6 ð6þ =L for the pre-heating zone between regions II and IV (F 0 and G 0 are the positions of the shocks on the axis of symmetry). Thus, we see that since the electron thermal conduction depends non-linearly on the temperature, the pre-heating zones have finite sizes. Therefore, we can expect (at least if the influence of the thermal conduction is not extremely strong) that the steady-state interaction of two gas sources will be characterized with preheating zones III 0 and IV 0 separated from the undisturbed radial flow zones by the surfaces P F and P G (Fig. 1). In the undisturbed regions equations ð3þ and ð4þ are still valid and determine the boundary conditions at H i and at the other boundaries of the calculation area. 3 NUMERICAL METHOD In our calculations we use the soft fitting technique developed by Godunov et al. (1976) on the basis of the Godunov scheme (Godunov 1959). The technique allows us to combine the advantages of the Eulerian and Lagrangian approaches. When coupled with the physical processes splitting method (Yanenko 1967), it also allows us to consider different physical processes by making use of various powerful tools which have been developed for studying the influence of the corresponding processes on the gas flow. The general structure of the scheme in the 1D case has been described in detail by Zhekov & Myasnikov (1998), and by Myasnikov & Zhekov (1997) in the 2D case. We note here that since we substantially use the a priori information of the solution structure (Section 2), two typical grid structures were used in our calculations (Fig. 2). 4 RESULTS OF NUMERICAL MODELLING As can be seen from equations ð1þ, ð3þ and ð4þ, the solution in general depends on six dimensionless parameters: L, x, G thc, g, K 1;2. Figure 2. A schematic structure of the grids used for calculating the cases with L 1 (upper panel) and L q 1 (lower panel). It was shown by Lebedev & Myasnikov (1988, 1990) that the adiabatic solution (G thc ¼ 0) does not depend on K 1;2 if the condition K 1;2 p 1 is fulfilled. It turns out that the same is true for the thermal conduction case. Since the importance of thermal conduction increases with the binary separation (see equation 2), i.e. its effect is more pronounced in wide binaries, we will not consider the dependence of the flow parameters on K 1;2 in the present study and their values will be kept fixed at K 1;2 ¼ 1 10 ¹4. We will also fix g ¼ 5=3, and will mainly study the dependence of the flow characteristics on L, x and G thc. In order to better illustrate the influence of thermal conduction on the CSW characteristics, we will first consider the case of two equal winds: L ¼ 1 and x ¼ 1. In this case, thermal conduction manifests itself mainly through the appearance of the pre-heating zones: the zones in which the hot interaction region I+II heats the cool free winds in regions III and IV. Figs 3 and 4 present the logarithm density (the upper panels) and the logarithm temperature (the lower panels) contours for the cases with G thc ¼ 100 and G thc ¼ 1000, respectively. We note that the positions of the shocks are found by the density jumps while the pre-heating zones are clearly seen in the temperature contours. Moreover, the wiggles in the density contours correspond to the positions of the thermal waves which determine the extension of the pre-heating zones. It is seen that the larger the value of G thc, the larger the size of the pre-heating zones. On the contrary, the size of the interaction region (I+II) becomes smaller with the increase of the values of the G thc coefficient. These

4 Dissipative models of colliding stellar winds I 689 Figure 3. Logarithm density (upper panel) and temperature (lower panel) contours for the case with L ¼ 1, x ¼ 1, G thc ¼ 100. Figure 4. Logarithm density (upper panel) and temperature (lower panel) contours for the case with L ¼ 1, x ¼ 1, G thc ¼ effects of thermal conduction are well understood, since the existence of the pre-heating zones suggests in general that the gas temperature in this part of the free winds is higher than it is in the adiabatic case. This also means that the Mach number in front of the shocks will be smaller in the conductive case than in the adiabatic one. As a result, the gas pressure behind the shocks will be smaller Figure 5. The (a) velocity, (b) density and (c) temperature distributions along the axis of symmetry for the cases with L ¼ 1, x ¼ 1 and G thc ¼ 0 (curves 1), G thc ¼ 100 (curves 2), G thc ¼ 1000 (curves 3). in the conductive case. Thus, the higher the values of G thc, the smaller the Mach number and the smaller the gas pressure in the interaction region as well. The consequence of this is that the shocks are moving further from the gas sources to find a new equilibrium position, i.e. the size of region I+II becomes smaller. On the other hand, the higher the values of G thc are, the closer the shocks are to isothermal shocks and therefore the higher the density jump at the shocks (Fig. 5b). Additionally, the increase of G thc results in a decrease in the value of the temperature inside the region I+II and in a smoother temperature distribution inside the zones (Fig. 5c). This is natural since thermal conduction simply redistributes the same amount of internal energy among the total (larger) amount of gas inside the interaction region and the pre-heating zones. A quite complicated flow structure appears in the interaction region I+II when thermal conduction is considered in the cases with x 1. It is known (Lebedev & Myasnikov 1988, 1990) that in the adiabatic case the geometrical structure of the flow does not depend on x, and the gas dynamical parameters can be obtained for different x by a simple renormalization rðxþ ¼rð1Þ=x 2 and vðxþ ¼vð1Þxin the regions II+IV. Thus, the influence of thermal conduction for the cases with x 1 can be studied in the following way. First, we obtain numerically the basic solution for the case with x ¼ 1 and G thc ¼ 0. Then, thanks to the fact that we know exactly the position of the softly fitted contact discontinuity, we renormalize the solution in the way described above. Finally, we

5 690 A. V. Myasnikov and S. A. Zhekov switch on thermal conduction with the value of G thc in which we are interested. This step-by-step procedure allows us to study not only the steady-state solution (which is of most interest here) but also the way in which the solution forms from the initial adiabatic one. This latter may be useful in order to understand some of the details in the steady-state solution. In order to study the dependence of the solution on x, wefix L¼1, G thc ¼ 1000 and vary x. Fig. 6 presents (a) the velocity, (b) the logarithm density and (c) the logarithm temperature distributions along the axis of symmetry for the cases with x ¼ 0:5; 0:1 (curves 2 and 3, respectively). For comparison, the case with x ¼ 1 is also shown in Fig. 6 (curve 1). In the case with x ¼ 0:5, the adiabatic temperature and the density jump at the contact discontinuity equals 4. This value is not very large, but the effect of thermal conduction at the contact discontinuity is noticeable for the chosen value of G thc ( G thc q 1). One can see, first, that thermal conduction equals the temperature values at the both sides of this discontinuity and the density values as well (Fig. 6b,c, curve 2). The position of the point where the velocity is equal to zero (i.e. the contact discontinuity position) is shifted towards the source with the higher flow velocity (this source is in the origin of the coordinate system) as compared with the position of this surface in the case x ¼ 1 (Fig. 6a, curves 1 and 2). That is because, when the thermal conduction is switched on and the hotter region I starts heating the colder region II, the pressure at Figure 6. The (a) velocity, (b) density and (c) temperature distributions along the axis of symmetry for the cases with L ¼ 1, G thc ¼ 1000 and x ¼ 1 (curves 1), x ¼ 0:5 (curves 2), x ¼ 0:1 (curves 3). the contact discontinuity increases in region II and at the same time it decreases in region I (one loses energy and the other gains it). This pushes the contact discontinuity towards the hotter gas until a new equilibrium pressure value is found. From a physical point of view, the heating of the cool gas leads to its mass evaporation from the initial contact discontinuity, which should therefore be considered just a surface from which the new outflow is realized. The result of the evaporation is the smoothing of the density distributions in the interaction region I+II (Fig. 6b). Secondly, the shift of the contact discontinuity towards the source having the higher flow velocity causes the shift of the shock F in the same direction. At the same time, however, the size of the region III 0 decreases with respect to its value in the case x ¼ 1. The last is naturally a result of the fact that a part of the energy in region I is lost in heating region II. Finally, since the temperature in regions II+IV is lower than it is in the case with x ¼ 1, the thermal flux at the shock G is smaller as well. Therefore, the pre-heating zone IV 0 has a smaller size than that of zone III 0 and the temperature distribution is steeper in region IV 0 (Fig. 6c). In the case with x ¼ 0:1, the effects of thermal conduction are, in general, the same as in the case with x ¼ 0:5. However, since the density and the temperature jumps at the contact discontinuity are equal to 100 when x ¼ 0:1, some additional effects take place. Notably, because of the high gradients, the thermal flux through the contact discontinuity is so large that the pressure increase in the vicinity of the contact discontinuity in region II is much greater than it is in the rest of this region when the thermal conduction is switched on. As a result, a pressure pulse at the contact discontinuity pushes the gas out of the initial contact discontinuity in both directions. The contact discontinuity moves towards the hotter gas as in the case where x ¼ 0:5, and the gas in region II decelerates. This effect is clearly seen in the non-monotonic velocity distribution (Fig. 6a) between the contact and the evaporation surfaces where a strong density rarefaction also takes place (Fig. 6b). Accepting the notation that region I is the region bounded by the shock F and the evaporation surface, one can see that in the case in which the thermal conduction is large enough, the main part of region I consists of the evaporated gas. That is in accordance with the basic assumption accepted by Weaver et al. (1977), who considered the effect of thermal conduction in a similar solution for the wind-blown bubbles, and also with the results of numerical modelling of the same flow performed by Zhekov & Myasnikov (1998). The evaporation effect can be seen very well for small x. For example, the temperature in region II+IV is so small in the case x ¼ 0:1 that the region IV 0 practically does not exist. One can see that the temperature has its normal jump on the shock G and then its profile is similar to the temperature profile in the pre-heating zone. This is so because of the initial large temperature difference between regions I and II. In fact, we noted that the contact discontinuity has moved towards the source with the higher flow velocity (see curve 3 in Fig. 6a), which is now not at Z ¼ 1asit should be in the case with L ¼ 1. Additionally, a large density gradient has appeared in region II (see curve 3 in Fig. 6b) which simply coincides with the steep temperature profile there (see curve 3 in Fig. 6c). Thus, the region between the contact discontinuity and this steep temperature/density gradient is occupied by the gas that has evaporated from region II. The size of this new preheating zone is smaller than the size of region III 0 and this is well understood: the density in region II is much higher than the density in region III when x ¼ 0:1. Then, it is clear from a physical point of view that media with a higher density are heated less efficiently than

6 Dissipative models of colliding stellar winds I 691 those with a lower density, since the same amount of thermal energy is spread over a different number of particles. An interesting point relating to the CSW with small x is the possibility of the appearance of some instabilities. As was noted by Stevens et al. (1992), interacting flows with x 1 and G thc ¼ 0 are subject to the Kelvin Helmholtz (KH) instability which takes place in the vicinity of the contact surface. Our calculations (Myasnikov, Belov & Zhekov 1997) confirmed this statement for the existence of the instability. We noticed, however, some properties of the unstable flows which contradict those pointed out by Stevens et al. (1992). On the other hand, our new results are in accordance with the predictions of the linear stability analysis, carried out recently by Belov (1997a c). Specifically, we found that the stagnation point is never itself unstable, and also that the instability is more pronounced in the plane case than in the axially symmetric one. Apart from the further discussion of this topic, which is interesting from the hydrodynamical point of view (see details in Myasnikov et al and Belov & Myasnikov 1998), we would like to point out the following. As we have seen, large values of G thc lead to a diminution of the jump in the gas dynamical parameters at the contact discontinuity, and it is natural to expect that these values of G thc may damp this instability. On the other hand, an interesting question arises, i.e. whether small values of G thc might damp the KH instability as well. To study this problem, we ran the same case of CSW that was presented in Myasnikov et al. (1997). The result of modelling this case with L ¼ 1, x ¼ 0:1 and G thc ¼ 0:1 is presented in Fig. 7 for the same instant of time as for the case in fig. 7 in Myasnikov et al. (1997). One can see that a narrow thermal boundary layer exists near the contact discontinuity and it is noticeable up to r 0:4. It has to be mentioned that this boundary layer is resolved in our calculations, that is, its size is larger than the distance between the grid points ( 4:8 10 ¹3 versus DzðrÞ ¼3 10 ¹3 at r 0:25). On the other hand, the solution is qualitatively similar to that presented in fig. 7 in Myasnikov et al. (1997) but the instability at r ¼ 0:25 (this is the point where it starts being noticeable in the adiabatic case) becomes less pronounced. Therefore, it is conclusive that the KH instability still exists for small values of G thc. Interestingly enough, one can note that the KH instability now takes place not at the contact discontinuity but around the evaporation surface instead. This might qualitatively explain why it is becoming less noticeable simply because the evaporation surface is smoother than the contact one in the sense that the profiles of the physical quantities are continuous and there are no jumps. We did not find the minimum value of G thc at which the instability disappears the calculations are much too time consuming. Moreover, we would like to mention that, on the one hand, such experiments are only of academic interest since the instability obtained does not have direct observational evidence. On the other hand, it is worthwhile doing such numerical experiments since the KH instability may provide seed perturbations for developing other kind of instabilities in the CSW. This means that in the flows with strong thermal conduction effects, the KH instability, and other instabilities as well, might be suppressed. Next, we will consider the dependence of the conductive CSW on L. The sizes of the pre-heating zones III 0 and IV 0 on the axis of symmetry are given in Table 1 for the case of G thc ¼ 1000 and x ¼ 1. For comparison, the values obtained with the help of formulae ð5þ and ð6þ are also presented. It is immediately seen that the size of the pre-heating zones decreases with the increase of L, and as a rule region III 0 is more extended than region IV 0. These particularities might be explained as follows. When L increases, the whole interaction region moves towards the source in the origin of the coordinate system. Since x does not change we have the same amount of internal energy which must be spread over the free wind, which means that the same amount of gas of the free wind must be heated. However, the density profile in the free wind is steeper in the region closer to the flow source, which results in a smaller size for the pre-heating zone. On the other hand, increasing L with fixed x, we are effectively increasing the mass loss of the second source (that is the one which is located at distance 2d from the origin of the coordinate system). In this case, there will be an increase of the density of the free wind in region IVand, since again the amount of energy to be spread over the wind is the same, the result is a decrease of the size of the pre-heating zone IV 0. Also, we would note that the numerically derived values for the sizes of the pre-heating zones are always smaller than those predicted by the analytical formulae. This discrepancy is readily explained since formulae ð5þ and ð6þ have been derived assuming that the temperature at the shocks was equal to the value given by the adiabatic shock jump conditions. In the self-consistent numerical model it is lower than this value, since part of the energy is used to heat the gas in front of the shocks. An interesting case is when L q 1, and for large enough values of this parameter the configuration presented in Fig. 8 may take place. Specifically, as a result of the high density in front of the shock F in the vicinity of the axis of symmetry, the pre-heating zone is narrower there in comparison with its size downstream where the Table 1. Figure 7. The logarithm density contours for the case with L ¼ 1, x ¼ 0:1, G thc ¼ 0:1. L Z F Z F Z G Z G (est.) (num.) (est.) (num.) 1 0: : :

7 692 A. V. Myasnikov and S. A. Zhekov Figure 8. The logarithm density (upper panel) and the logarithm temperature (lower panel) contours for the case with L ¼ 30, x ¼ 1, G thc ¼ density of the free wind is lower. If G thc is large enough, the region III can be preheated up to the axis of symmetry. The larger the G thc coefficient, the smaller the size of the undisturbed part of region III (Fig. 9a). At the same time, in spite of the fact that the boundary P F of the pre-heating zone III 0 can be noticed at the density contour plot as well (Fig. 8, upper half), the density distribution is slightly affected by the thermal conduction in the tail part of the interaction region (Fig. 9b). It is interesting to consider the effects of the thermal conduction on the emission (observational) properties of the interaction region. To do this we model a binary system with the following basic parameters: Ṁ 1 ¼ 2 10 ¹6 M yr ¹1, V 1 ¼ 2000 km s ¹1, d ¼ cm, L ¼ 30, x ¼ 1 and G thc ¼ 0, 10 3 and 10 4, respectively. We note that the basic physical parameters are close to those of the WRþO binary WR140 but the chemical composition is solar. The corresponding X-ray spectra are presented in Fig. 10. One can see that the main effect on the X-ray emission is an increase of the luminosity with respect to the adiabatic case (G thc¼0 ) which readily follows from the numerical results presented above. It was mentioned that, on the one hand, thermal conduction causes an increase of the density in the interaction region and, on the other hand, the size of the region occupied by the dense gas becomes smaller. However, since the total emission of the optically thin plasma is proportional r 2 V (where r is the density and V is the volume), we can qualitatively expect an increase of the luminosity. Also, the thermal conduction lowers the Figure 9. The logarithm (a) temperature and (b) density distribution at the axis of symmetry for the case with L ¼ 30, x ¼ 1. Curve 1 corresponds to G thc ¼ 2000, 2 G thc ¼ 5000, 3 G thc ¼ temperature in the interaction region and this is why the X-ray spectra are becoming softer with increasing G thc. This effect is better seen in the spectral region at energies above 4 kev (Fig. 10). Finally, we would like to recall that thermal conduction effects will be important not only in wide binary systems but also in those systems the components of which possess fast enough winds. This is because of the strong dependence of the conductivity on the wind velocity (see equation 2). If we consider WRþO binaries, WR140 is an example of a system where these effects should be taken into account. Other objects where the CSW model could in principle include similar physics are those WRþO wide binaries, similar to WR140 mainly by the so-called periodic dust formation in their winds (e.g Williams 1995). We note, however, that some of these objects show non-thermal radio emission which might be suggestive of the importance of magnetic fields in their CSW physics. Mainly because of the geometrical complexity of the magnetic field thermal conduction interaction, we postpone such studies to a future work. As to the PMS binaries, one can expect that the thermal conduction may play some role only in systems with binary separation d cm and mass loss Ṁ 10 ¹7 M yr ¹1, mainly because of their relatively slow winds of a few hundred km s ¹1. 5 CONCLUSIONS We have considered a numerical 2D model of colliding stellar winds in wide binary systems which self-consistently takes into

8 Dissipative models of colliding stellar winds I 693 region if the thermal conductivity is large enough. As a result, a so called evaporation zone appears in the interaction region. (iii) Small values of the thermal conductivity do not damp the Kelvin Helmholtz instability if it exists in the adiabatic CSW. However, in the conductive CSW the KH instability develops around the evaporation surface instead of at the contact one. Further studies in this direction might be of great interest, especially if they are combined with an analytical investigation of the problem. (iv) The thermal conduction may also have an effect on the emission properties of the interaction region. Particularly, including this process in the CSW physics results in an increase of the X-ray emission in comparison with the adiabatic case. The X-ray luminosity ( kev) increases and the X-ray spectrum becomes softer. Figure 10. The X-ray spectra ( kev) for the case of a binary system with basic parameters: Ṁ 1 ¼ 2 10 ¹6 M yr ¹1, V 1 ¼ 2000 km s ¹1, d ¼ cm, L ¼ 30, x ¼ 1 and G thc ¼ 0 (solid line), 10 3 (dotted line) and 10 4 (dashed line), respectively. The X-ray absorption by the stellar winds is taken into account (more massive wind is in front in the lower panel). account the electron thermal conduction. The model is based on the use of the soft fitting Godunov technique coupled with the physical splitting method. The main results of the study are as follows. (i) Pre-heating zones in the cool free-flowing winds appear when the electron thermal conduction efficiently influences the CSW. In these zones the gas temperature increases in front of the shocks which lowers the Mach number and, therefore, the density jump at the shocks. The pre-heating zones have finite sizes, which is a result of the non-linear dependence of the thermal flux on temperature. The numerically derived values of the sizes on the axis of symmetry are compatible with those estimated by simple formulae derived in the present study. (ii) The electron thermal conduction leads to a significant redistribution of the gas dynamical parameters in the interaction ACKNOWLEDGMENTS AVM thanks JSPS and the Monbusho Grant-in-Aid for Encouragement of Young Scientists (No ) for providing the opportunity to initiate this work and the Russian Foundation of Basic Research for financial support under Grant No The authors acknowledge the ESO C&EE Program Travel Grant No. A Partial financial support was also supplied by contract F-570 with the National Science Fund of the Bulgarian Ministry of Education, Science and Technologies. The authors are grateful to the referee Prof. J. Dyson for criticism and valuable suggestions. REFERENCES Belov N. A., 1997a, Fluid Dyn., 32, 219 Belov N. A., 1997b, Fluid Dyn., 32, 728 Belov N. A., 1997c, J. Fluid Mech., submitted (Institute for Problems in Mechanics, Moscow, Preprint No. 588) Belov N. A., Myasnikov A. V., 1998, Fluid Dyn., in press Conti P. S., 1988, in Conti P. S., Underhill A., eds, O stars and WR stars. NASA SP-497, 81 Godunov S. K., 1959, Mat. Sbornik, 47, 271 Godunov S. K., Zabrodin A. V., Ivanov M.Ya, Kraiko A. N., Prokopov G. P., 1976, Chislennoe Reshenie Mnogomernih Zadach Gazovoi Dinamiki. Nauka Press, Moscow (French translation in: Multidimensionnels de la Dynamique des Gaz, Moscow, Editions MIR, 1979) Kallrath J., 1991, A&A, 247, 434 Lebedev M. G., Myasnikov A. V., 1988, in Paskonov V. M., Roslyakov G. S., eds, Numerical Methods in Aerodynamics. Moscow State University Press, Moscow, p. 3 Lebedev M. G., Myasnikov A. V., 1990, Fluid Dyn., 25, 629 Leinert Ch., Zinnecker H., Weitzel N., Christou J., Ridgway S. T., Jameson R., Haas M., Lenzen R., 1993, A&A, 278, 129 Luo D., McCray R., MacLow M.-M., 1990, ApJ, 362, 267 Myasnikov A. V., Zhekov S. A., 1991, Ap&SS, 184, 287 Myasnikov A. V., Zhekov S. A., 1993, MNRAS, 260, 221 Myasnikov A. V., Zhekov S. A., 1997, Institute for Problems in Mechanics, Moscow, Preprint No. 595 Myasnikov A. V., Belov N. A., Zhekov S. A., 1997, Institute for Problems in Mechanics, Moscow, Preprint No. 582 Myasnikov A. V., Zhekov S. A., Belov N. A., 1998, MNRAS, 298, 1021 Pollock A. M. T., 1987, ApJ, 320, 283 Preibisch Th., Zinnecker H., 1994, A&A, 292, 152 Prinja R. K., Barlow M. J., Howarth I. D., 1990, ApJ, 361, 607 Spitzer L., Jr., 1962, Physics of Fully Ionized Gasses. Interscience Press, New York Stevens I. R., Blondin J. M., Pollock A. M. T., 1992, ApJ, 386, 265 Stevens I. R., Corcoran M. F., Willis A. J., Skinner S. L., Pollock A. M. T., Nagase F., Koyama K., 1996, MNRAS, 283, 589

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