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Astron. Astrophys. 357, 180-196 (2000)

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3. Numerical simulations of the formation of detached, geometrically thin gas shells

In the following we present the results of a number of numerical simulations that were carried out in an effort to investigate the hydrodynamical details regarding the two competing scenarios currently being discussed as possible explanations for the existence of thin circumstellar CO shells. The goal is to find out whether both scenarios are viable from a hydrodynamical point of view, and whether we can find any clues as to what is the most likely explanation: mass loss `eruption' , two-wind interaction , or a combination of both .

3.1. Mass loss `eruption' models

Case A : the S Scuti scenario

In order to explain the observed IR spectral energy distribution of the well-known carbon star S Sct, Groenewegen & de Jong (1994) argued that a mass loss `eruption' must have taken place about 10 000 yrs ago. Based on their dust radiative transfer models they conclude that the `eruption' lasted for about 1000 years at a rate of [FORMULA]. Their favored mass loss history is indicated by the dashed line in Fig. 1.

[FIGURE] Fig. 1. Top: Two different mass loss `eruption' models investigated in this work: the mass loss history suggested by the properties of the CO shell of TT Cyg (Case B , solid), compared to the mass loss history proposed by Groenewegen & de Jong (1994) for S Sct (Case A , dashed). For the times labeled `1' to `10', the radial distribution of velocity and density is shown in Fig. 2. Middle: Gas velocity u (solid) and dust velocity w (dotted) as a function of time at a radial distance of [FORMULA] cm, as obtained from two-component radiation hydrodynamics for the TT Cyg mass loss `eruption' model shown above. Bottom: Dust temperature as a function of time at radial distances of [FORMULA] cm (upper) and [FORMULA] cm (lower) for the TT Cyg mass loss `eruption' model, as obtained by DEXCEL from the condition of dust radiative equilibrium.

Several remarks are in order. First, we point out that Groenewegen & de Jong ignore the hydrodynamics of the problem. Instead, they adopt a simple kinematic approach, assuming a uniform wind velocity of 16.5 km s-1, irrespective of the mass loss rate. According to our radiation hydrodynamics calculations, this is not a valid approximation (see below). Second, in order to explain the dust emission at [FORMULA] 100 µm, they have to assume a rather high dust-to-gas ratio of [FORMULA]. We have recalculated their case with our two-component radiation hydrodynamics code, adopting the same stellar parameters, mass loss history, dust properties and dust-to-gas ratio, and find that the resulting gas outflow velocity is as high as 80 km s-1 before the `eruption' (which is clearly unrealistic) and about 10 km s-1 afterwards (demonstrating that the outflow velocity is actually not independent of the mass loss rate). Even though the radial dust distribution is distinctly different in the kinematic and the hydrodynamical case, both models contain the same total amount of dust and perfectly fit the observed IR spectral energy distribution of S Sct (not shown). Interestingly, the hydrodynamical model calculations based on stellar evolution calculations with mass loss can also explain the IR emission of S Sct (cf. Fig 18 of SSS98), although the underlying mass loss history is completely different (top panel of Fig. 7 below).

Finally, we have recalculated the case of Groenewegen & de Jong with a more realistic dust-to-gas ratio of [FORMULA], and find a gas outflow velocity of 19 km s-1 before the `eruption' (which is much more reasonable) and of 6 km s-1 afterwards. This model fails, however, to account for the dust emission observed at [FORMULA] 100 µm.

In summary, we see no convincing arguments in favor of the mass loss `eruption' scenario as far as the modeling of the dust component is concerned.

Case B : the TT Cygni scenario

Even though no physical mechanism is presently known to produce very short mass loss `eruptions' of the required strength (see however recent claims by Schröder et al. 1998), it is nevertheless tempting to relate the existence of the observed very thin CO shells to this presumed phenomenon. According to Olofsson et al. 1998, the CO shell of TT Cyg has a width of [FORMULA] cm and a radius of [FORMULA] cm. Assuming a constant expansion velocity of 13 km s-1, this implies a mass loss `eruption' that happened about 12 000 yrs ago and lasted for about 500 yrs at a rate of [FORMULA] (estimated shell mass [FORMULA]). We have constructed a schematic mass loss history in accordance with the above mentioned data for TT Cyg (Fig. 1, solid line) and used it as a boundary condition for a time-dependent hydrodynamical simulation of the mass loss `eruption' scenario.

First we applied the DEXCEL code to this problem. The results are displayed in the lower panels of Fig. 1 and in the left-hand column of Fig. 2. Before the `eruption', the gas outflow velocity is [FORMULA] km s-1, well within the observed range for carbon-rich AGB stars. The drift velocity of the dust relative to the gas is considerable, [FORMULA] km s-1. During the `eruption', the gas outflow velocity is slightly reduced due to the increased optical depth of the heavy outflow, [FORMULA] km s-1, while the drift velocity becomes essentially zero. After the `eruption', the density of the wind is very low, so gas and dust decouple, as indicated by the high drift velocity, [FORMULA] km s-1. At the same time the gas velocity has declined to [FORMULA] km s-1, in agreement with observational evidence for TT Cyg.

[FIGURE] Fig. 2a-d. Evolution of a shell of enhanced gas density produced by the TT Cyg mass loss `eruption' scenario defined in Fig. 1. Top to bottom: Radial distribution of a gas outflow velocity, b gas density, c differential shell mass [FORMULA] (symbols mark the radial grid points; `measurements' of the half widths, [FORMULA], indicated by vertical and horizontal bars), and d relative shell width [FORMULA] (dashed curve [FORMULA] corresponding to [FORMULA] const.). The labeled curves in the upper three panels correspond to times `1' to `10' indicated in Fig. 1. The left-hand column displays the results obtained with the DEXCEL two-component radiation hydrodynamics code (assuming [FORMULA], [FORMULA] K, [FORMULA]), while the right-hand column shows the corresponding time sequence as obtained with the NEBEL one-component hydrodynamics code for the same [FORMULA], [FORMULA], [FORMULA] at the inner boundary at [FORMULA] cm. Different line styles serve to distinguish the curves [FORMULA] and [FORMULA] at different times.

The gas shell produced by the mass loss `eruption' moves outward at an average speed of [FORMULA] km s-1 (which in this case practically coincides with the flow velocity at the center of the shell, [FORMULA]), somewhat higher than the Doppler velocity measured in the CO shell of TT Cyg. However, the substantial velocity gradient across the shell that develops after some time leads to a considerable broadening of the density structure as it recedes from the star, and to a corresponding decline of its amplitude (shell mass [FORMULA] const.). Beyond [FORMULA] cm, the relative thickness of the shell remains approximately constant at a value of [FORMULA].

We have recalculated this test case with the NEBEL code, using the DEXCEL results at [FORMULA] cm (partly shown in the lower panels of Fig. 1) as a time-dependent inner boundary condition. The evolution of the ejected shell is shown in the right-hand column of Fig. 2, while Fig. 3 displays details of the structure as it appears 5 867.5 yrs after the end of the `eruption'.

[FIGURE] Fig. 3. Radial distribution of gas velocity (top), particle number density (middle) as well as gas temperature and pressure (bottom, solid / dashed) at time [FORMULA] yrs (between points 8 and 9), as obtained with the NEBEL one-component hydrodynamics code for the TT Cyg mass loss `eruption' scenario defined in Fig. 1. By this time the velocity plateau in the central part of the density peak has disappeared. Symbols (+) indicate the numerical grid.

The propagation velocity of the gas shell produced by the mass loss event is only marginally smaller than that found with DEXCEL, [FORMULA] km s-1. However, the extended velocity plateau in the central part of the density peak survives much longer than indicated by the DEXCEL results. As long as it exists, the absolute shell width (FWHM) remains approximately constant: [FORMULA] cm, corresponding to the original width of the `eruption', [FORMULA]. When the shell has moved out beyond [FORMULA] cm, the velocity plateau has finally dissolved due to the presence of thermal pressure gradients. The resulting positive mean velocity gradient then leads to some noticeable broadening, such that the relative thickness of the shell remains approximately constant at [FORMULA].

In another NEBEL run, we have repeated this test problem without any source terms in the energy equation. In this case, the temperature drops far below the dust radiative equilibrium temperature due to adiabatic cooling of the expanding outflow. As a consequence of the concomitant overall reduction of the gas pressure, the average outflow velocities are slightly reduced, and [FORMULA] km s-1. The velocity plateau proves to be very stable in this situation, and the absolute shell width remains essentially constant at [FORMULA] cm. This result demonstrates that the broadening found in the above case where [FORMULA] is indeed caused by the thermal pressure and not by numerical diffusion.

Clearly, the initial velocity plateau is the the result of the constant outflow velocity during the mass ejection, [FORMULA] km s-1. The plateau is eroded by rarefaction waves running into the density peak from the edges. Hence, the life time of the velocity plateau is [FORMULA], where [FORMULA] is the local sound speed. For the non-adiabatic case, [FORMULA] km s-1, and [FORMULA] yrs, in close agreement wit the NEBEL result. This estimate also explains the NEBEL results for the adiabatic case, where [FORMULA] is at least a factor of 3 smaller.

We have to conclude that the strong dispersion of the density structure as found with DEXCEL is an artifact, caused by substantial numerical diffusion. The results obtained with NEBEL are much more reliable, as expected for a second-order Godunov-type scheme. Still, DEXCEL is needed to compute the wind acceleration in the inner region of the dusty envelope.

Case C : modified TT Cygni scenario

For the sake of completeness, we have investigated a related mass loss `eruption' model, which is obtained by reversing the mass loss rates before and after the `eruption' in the TT Cyg scenario studied before. In this case, called the modified TT Cyg scenario in the following, we have a slow wind before, and a faster wind after the mass loss event (upper panel of Fig. 4). There is no known observational counterpart for this situation as far as AGB stars are concerned, but it may be realized in other astrophysical environments. Anyway, it is interesting to see whether the conclusions drawn above remain valid also under these circumstances.

[FIGURE] Fig. 4. Modified mass loss `eruption' model (Case C ) obtained by reversing the mass loss rates before and after the `eruption' in the original TT Cyg mass loss history (Fig. 1). Gas and dust velocity at [FORMULA] cm (middle) as well as temperature (at [FORMULA] and [FORMULA] cm; bottom) as a function of time are obtained from two-component radiation hydrodynamics for the mass loss `eruption' model shown in the top panel. For the times labeled `1' to `12', the radial distribution of velocity and density is shown in Fig. 5.

As before, we applied the DEXCEL code first. The results are displayed in the lower panels of Fig. 4 and in the left-hand column of Fig. 5. By design, the conditions before and after the `eruption' are now reversed: initially, the density of the wind is very low, and so is the gas outflow velocity, [FORMULA] km s-1, while the drift velocity is high, [FORMULA] km s-1. During the `eruption', the gas velocity jumps up to [FORMULA] km s-1, as the gas/dust coupling becomes very efficient and the drift velocity has strongly declined to almost zero. After the event, the gas velocity increases slightly to [FORMULA] km s-1, since the spectral energy distribution is blue-shifted with decreasing optical depth of the outflow, leading to a somewhat enhanced radiation pressure on dust.

[FIGURE] Fig. 5a-d. Same as Fig. 2, but for the modified TT Cyg mass loss `eruption' scenario defined in Fig. 4. The labeled curves in the upper three panels correspond to times `1' to `12' marked in Fig. 4. The left-hand column displays the results obtained with the DEXCEL two-component radiation hydrodynamics code while the right-hand column shows the corresponding time sequence as obtained with the NEBEL one-component hydrodynamics code using the same [FORMULA], [FORMULA], [FORMULA] at the inner boundary at [FORMULA] cm as input.

The gas shell produced by the `eruption' moves outward at an average speed of [FORMULA] km s-1, the flow velocity at the center of the shell. Note that this value is significantly lower than found in the original TT Cyg mass loss eruption scenario ([FORMULA] km s-1). As before, the initial velocity plateau in the central parts of the density peak disappears quickly and gives way to a substantial velocity gradient across the shell, this time of negative sign. Under these circumstances the shell is compressed and initially is even narrower than [FORMULA]. Over the whole radial range, the relative thickness of the shell remains approximately constant at a rather low value of [FORMULA]. It turns out, however, that this result is again a numerical artifact. Curiously, numerical diffusion produces a too narrow density structure in this case.

This becomes clear when recalculating this test case with the NEBEL code, again using the DEXCEL results at [FORMULA] cm (partly shown in the lower panels of Fig. 4) as a boundary condition. The NEBEL results are shown in the right-hand column of Fig. 5, and in Fig. 6.

[FIGURE] Fig. 6. Radial distribution of gas velocity (top), particle number density (middle) as well as gas temperature and pressure (bottom, solid / dashed) at time [FORMULA] yrs (between points 8 and 9), as obtained with the NEBEL one-component hydrodynamics code for the modified TT Cyg mass loss `eruption' scenario defined in Fig. 4. Symbols (+) indicate the numerical grid.

The propagation velocity of the gas shell produced by the mass loss event is distinctly larger than that found with DEXCEL, [FORMULA] km s-1. This is very nearly the same propagation velocity as found before with NEBEL in the original TT Cyg mass loss `eruption' scenario (compare Figs. 3 and 6 which show the position of the shell at the same time, 5 867.5 yrs after the end of the mass loss `eruption'). The width of the shell as a function of radial distance is almost indistinguishable from the original test case. Obviously, the central parts of the density peak are not affected by wind interaction; the width is determined by the duration and outflow velocity of the `eruption'. Note that in contrast to the DEXCEL results, the action of the thermal pressure produces a positive velocity gradient at the center of the shell, which has the same size as in the original `eruption' scenario (compare the top frames of Figs. 3 and 6).

The wings of the density structure are certainly affected by the wind properties in front and behind. The reversed wind velocities now lead to the formation of both a forward and a reverse shock (see Figs. 5 and 6). The forward shock is stronger because the velocity jump is larger, and there is a substantial heating associated with this shock. But the heated region is confined to a thin layer in the outermost wing where the temperature is as high as 800 K. Radiative cooling is efficient enough to keep the temperature at a low constant level ([FORMULA] K) almost everywhere, and the density jump across the shock by far exceeds a factor of 4 (which is an upper limit in the "adiabatic" case).

Another NEBEL run without source terms in the energy equation exhibits the expected behavior: slightly lower outflow velocities, [FORMULA] km s-1, and essentially the same shell width, but now almost perfectly constant with radial distance. Heating in the leading shock produces post-shock temperatures in excess of 6 000 K in the absence of any radiative cooling. But again the high temperatures are confined to the regions of very low density in the outer wing of the peak.

Summarizing the results presented in this section, the simulations leave little doubt that it is indeed possible to produce rather narrow gas shells of enhanced density through a mass loss event of sufficiently short duration [FORMULA], as long as the outflow velocity is constant during the `eruption' and the stellar wind is highly supersonic. In order to explain the observed CO emission around TT Cyg, the requirement is [FORMULA] yrs. However, note that our simulations also indicate that the amount of dust ejected by such an `eruption' is insufficient to account for the observed excess dust emission at [FORMULA] 100 µm, assuming a realistic dust-to-gas ratio of [FORMULA] (cf. e.g. Kerschbaum & Hron 1996for the observed spectral energy distribution of TT Cyg).

3.2. Interacting winds

Interestingly enough, our simulations provide another mechanism creating thin shells of enhanced gas density, namely the interaction of two winds of different velocity and density. While analyzing DEXCEL simulations of the dynamical response of a carbon-rich circumstellar wind shell to the substantial temporal variations of stellar luminosity and mass loss rate associated with the final thermal pulses near the end of the AGB evolution, we discovered that the wind velocity is an almost bi-modal function of the mass loss rate, as described in more detail in SSS98.

Briefly, once a critical mass loss rate is exceeded, the coupling between dust and gas becomes efficient and causes a sudden transition from a slower, dense, "shock-driven" wind to a faster, less dense, "dust-driven" wind. Both type of winds carry approximately the same mass loss rate ([FORMULA] const.). This behavior inevitably leads to two-wind interaction : The faster wind in the inner parts of the shell runs into the slower wind in the outer parts, acting like a snow plow, piling up matter into a thin shell at the interface between the different kind of winds.

This mechanism is illustrated in more detail in Figs. 7 and 8. The top panel of Fig. 7 shows the behavior of the mass loss rate during a typical thermal pulse cycle. Recovering from the mass loss `interruption' near [FORMULA] yrs, the model's critical mass loss rate ([FORMULA]) is exceeded at [FORMULA] yrs, as indicated by the abrupt increase of the gas and dust velocity at this time (middle panel of Fig. 7). According to two-component radiation hydrodynamics simulations with DEXCEL, the gas velocity `jumps' up by a factor of 3, from 4 to 12 km s-1, within a time interval of 10 000 yrs. The reduced density in the inner parts of the dusty envelope leads to less absorption of stellar radiation and hence to a slight increase of the dust temperature in the outer regions beyond [FORMULA] cm (lower panel of Fig. 7).

[FIGURE] Fig. 7. Top: Mass loss rate as a function of time during a thermal pulse on the upper AGB. Data from stellar evolution calculations with mass loss by Blöcker (1995), for a solar composition star with an initial mass of [FORMULA] ending up as a white dwarf with [FORMULA]. During the shown time interval, the current stellar mass is reduced from 2.50 to [FORMULA]. The next thermal pulse starts at [FORMULA] yrs; [FORMULA] corresponds to the end of the AGB. Middle: Gas (solid) and dust (dotted) velocity as a function of time at a radial distance of [FORMULA] cm as obtained from two-component radiation hydrodynamics calculations based on the stellar evolution sequence shown above. The plotted time interval corresponds to the phase of steadily increasing mass loss rate between two thermal pulses. For the times labeled `1' to `11', the radial distribution of velocity and density is shown in Fig. 8. Bottom: Dust temperature as a function of time at radial distances of [FORMULA] cm (upper) and [FORMULA] cm (lower) as obtained from two-component radiation hydrodynamics for the same case.

[FIGURE] Fig. 8a-d. Same as Fig. 2, but showing the creation and evolution of a shell of enhanced gas density due to two-wind interaction . In each of the upper three panels, there are 11 different curves corresponding to times labeled `1' to `11' in Fig. 7. Data taken from the standard carbon star sequence described in SSS98. The results of the NEBEL one-component hydrodynamics code (shown in the right-hand panels) were obtained by using [FORMULA], [FORMULA], [FORMULA] from the DEXCEL two-component radiation hydrodynamics calculations (shown in the left-hand panels) at [FORMULA] cm as an inner boundary condition for NEBEL.

Fig. 8 shows the creation, evolution and propagation of a gas shell produced by two-wind interaction. Both the DEXCEL (left-hand column) and the NEBEL (right-hand column) simulation clearly demonstrate that the resulting density structure is a stable feature traveling to the outer regions of the circumstellar envelope without loss of identity. The mass of the shell increases as it keeps sweeping up matter on its way out. As expected, the results of the two codes differ quantitatively.

According to DEXCEL, the relative width of the shell is approximately constant beyond [FORMULA] cm, at a width-to-radius ratio of [FORMULA]. This is very similar to the result obtained for the mass loss `eruption' scenario C studied above, where we also found a negative velocity gradient across the shell, without any indication of a velocity plateau. In contrast, however, the amplitude of the quantity [FORMULA] ([FORMULA]: peak density of the shell) is not decreasing here with distance but remains approximately constant, indicating that the mass of the shell grows in proportion to its radial distance, [FORMULA]. More quantitatively, we measure a shell mass (integral over the range [FORMULA] [FORMULA] FWHM) of [FORMULA], a peak density of [FORMULA] g cm-3, and a density enhancement relative to the `background' of [FORMULA] at [FORMULA] cm, and [FORMULA], [FORMULA] g cm-3, [FORMULA] at [FORMULA] cm.

A closer inspection of the data reveals that the propagation velocity of the shell is slightly accelerated, increasing from [FORMULA]=6.6 km s-1 at [FORMULA] cm to [FORMULA]=8.6 km s-1 at [FORMULA] cm. We attribute this acceleration to the temporal increase of the speed of the inner wind (cf. middle panel of Fig. 7) and to the fact that the density structure of the outer envelope does not correspond to a constant mass loss rate ([FORMULA]) but exhibits a somewhat steeper gradient due to the steady growth of the stellar mass loss rate (factor of 10 over 30 000 yrs, see upper panel of Fig. 7).

Clearly, these DEXCEL results are affected by numerical limitations, viz. numerical diffusion (insufficient spatial resolution, especially in the outer parts of the model) and high numerical viscosity (in regions of local compression). Application of the NEBEL code to the same problem shows that the compressed gas shell is actually expected to be even narrower. The NEBEL results are shown in the right-hand column of Fig. 8, and in Fig. 9.

[FIGURE] Fig. 9. Radial distribution of gas velocity (top), particle number density (middle) as well as gas temperature and pressure (bottom, solid / dashed) at time [FORMULA] yrs (close to point 9), as obtained with the NEBEL one-component hydrodynamics code for the two-wind interaction scenario based on the stellar evolution sequence with mass loss shown in Fig. 7. Symbols (+) indicate the numerical grid. At this time the parameters of the AGB star are [FORMULA] K, [FORMULA], and the gas shell properties are [FORMULA], and [FORMULA].

In contrast to DEXCEL, the NEBEL simulations suggest that the relative shell width varies with radius as [FORMULA] (lower right panel of Fig. 8). In general, NEBEL produces even more narrow shells. At [FORMULA] cm, for example, the width-to-radius ratio is [FORMULA], at [FORMULA] cm [FORMULA] is less than 0.03 (see also Fig. 9). This is narrower than the shell widths obtained for all mass loss `eruption' scenarios studied previously.

As before, we find the propagation velocity of the shell to be slightly accelerated, increasing from [FORMULA] km s-1 at [FORMULA] cm to [FORMULA] km s-1 at [FORMULA] cm. The average shell propagation velocity is somewhat higher than indicated by DEXCEL. However, unlike DEXCEL, NEBEL reveals a velocity plateau at the center of the shell, indicating the development of a forward and a reverse shock of roughly equal strength (upper panel of Fig. 9).

The amplitude of the quantity [FORMULA] is now even increasing with distance; roughly [FORMULA]. This behavior is related to the fact that the velocity jump across the bounding shock fronts increases with radial distance, for the reasons mentioned above ([FORMULA]). Combined with the observation that [FORMULA] const., this again results in a shell mass that is proportional to the radial distance from the star, [FORMULA]. In this respect, the results from DEXCEL and NEBEL are consistent.

Summarizing, the results of our simulations covering the phase of steadily increasing mass loss after a helium-shell flash, strongly suggest that the mechanism of two-wind interaction is capable of producing very narrow circumstellar shells of compressed gas ([FORMULA]) without the need to invoke a mass loss `eruption' of very short duration. In the presence of molecular radiative cooling, shock heating of the shell material is negligible for the parameters considered here (see lower panel of Fig. 9)

In the following section we present a simple analytical model of two-wind interaction that is suitable to estimate the basic properties of the resulting shell, given the parameters of the two winds.

3.2.1. A simple analytical model of two-wind interaction

Consider two different, spherically expanding stellar winds: a "fast" inner wind characterized by a constant mass loss rate [FORMULA] and constant velocity [FORMULA], and a "slow" outer wind with parameters [FORMULA] and [FORMULA]. It can be shown that the amount of mass swept up by the "fast" inner wind interacting with the "slow" outer wind is given by the relation

[EQUATION]

where [FORMULA] is the distance of the interaction region from the central star, which increases proportional with time t (cf. Kwok et al. 1978, Kwok 1983). Since the situation encountered in the numerical simulations described in the preceding section may be approximated by [FORMULA], we simply have

[EQUATION]

Substituting typical values from our simulations ([FORMULA], [FORMULA] km s-1, [FORMULA] km s-1, [FORMULA] cm; cf. Figs. 7, 8), we find [FORMULA]. This number is in close agreement with the numerical results from both DEXCEL and NEBEL.

Assuming the collision of the winds to be completely inelastic (excess kinetic energy is instantly radiated away), the (constant) propagation velocity of the shell, [FORMULA], is given by

[EQUATION]

(cf. Kwok et al. 1978, Kwok 1983). If the wind densities are equal, then [FORMULA] and the shell propagation velocity is the straight mean of the two wind velocities:

[EQUATION]

For equal mass loss rates , Eq. (5) reduces to the simple result that the shell propagation velocity is the geometric mean of the two wind velocities:

[EQUATION]

Substituting the numerical values given above, we obtain [FORMULA] km s-1, while in the simulations we "measure" a shell velocity of about 7.5 km s-1 (DEXCEL) and 7.9 km s-1 (NEBEL), respectively, at [FORMULA] cm, in general agreement with the analytic estimate. As mentioned before, both simulations show an accelerated shell velocity, which is related to the fact that the interacting winds are not really constant, and therefore the assumption [FORMULA] is not strictly valid. In a simple numerical experiment, we have checked that NEBEL indeed gives a constant [FORMULA] of the correct magnitude if [FORMULA], [FORMULA], [FORMULA], [FORMULA] are fixed.

Further, Kwok et al. (1978) argue that the width of the shell is determined by the balance between the internal gas pressure of the shell, [FORMULA], and the external gas pressure plus ram pressure of the wind, [FORMULA]. With [FORMULA], [FORMULA], and [FORMULA], the relative width of the compressed gas shell can be estimated as:

[EQUATION]

where [FORMULA] is the isothermal sound speed. For [FORMULA], we obtain

[EQUATION]

In this approximation, the relative shell width is independent of the mass loss rate. Low gas temperature and large velocity difference ([FORMULA]) favor thin shells (Fig. 10). Assuming [FORMULA] K, [FORMULA], adopting the numerical values of [FORMULA] and [FORMULA] given above, and making use of Eq. (7), we obtain a relative width of 0.05, in reasonable agreement with what is found in the numerical simulations near [FORMULA] cm. According to Eqs. (8) or (9), we should expect a slight decrease of [FORMULA] with radial distance since [FORMULA]. This is not seen in the DEXCEL results, but in the NEBEL results, where, however, [FORMULA]. This steeper decline can be explained by the fact that both the density of the inner wind (the term [FORMULA]) and the velocity of the inner wind (the term [FORMULA]) increase with time, leading to an additional reduction of the shell width (see Eq. (8)). We have checked that NEBEL gives constant [FORMULA] in the idealized case of two interacting winds with fixed properties and constant temperature.

[FIGURE] Fig. 10. Relative shell width according to Eq. (9). Top : [FORMULA] as a function of [FORMULA] for fixed [FORMULA] km s-1, for [FORMULA] (dotted), 100 (solid) and 200 K (dashed). Bottom : [FORMULA] as a function of temperature for fixed [FORMULA] km s-1 and [FORMULA] km s-1, assuming a molecular weight of [FORMULA].

In the framework of the simple analytical model, we can finally estimate the ratio of gas density in the compressed shell to gas density in the smooth `background' flow (making use of Eq. (4)) as

[EQUATION]

which amounts to a factor of 30 in the above example, in close agreement with the NEBEL results shown in Fig. 9.

In summary, we conclude that the numerical results of the two-wind interaction process presented in the preceding section can largely be understood with the simple analytical model outlined above. The quantitative comparison of numerical and analytical results again indicates that the NEBEL code is more reliable than the DEXCEL code.

3.3. Modeling the circumstellar signature of a typical helium-shell flash

In Sect. 3.2 we have studied the dynamical behavior of the system AGB star + circumstellar envelope during the phase of steadily increasing mass loss after the luminosity minimum resulting from the preceding helium-shell flash. Now we are interested in the consequences of the shell flash itself. As before, this investigation is based on DEXCEL simulations of the dynamical evolution of a carbon-rich circumstellar wind shell of a thermally pulsing AGB star, as described in SSS98. For reasons to be explained later, we did not use the flash shown in Fig. 7 but the preceding one occurring near [FORMULA] yrs. Fig. 11 shows the temporal variation of luminosity and mass loss rate during this helium-shell flash. Starting from [FORMULA], the stellar luminosity begins to decline slowly, before dropping to a sharp minimum of [FORMULA]. Then it takes only 300 yrs for the luminosity to jump up to [FORMULA], followed by a more gradual decline to an extended, deep minimum centered at [FORMULA] yrs where [FORMULA]. According to the mass loss prescription by Blöcker (1995), the mass loss rate behaves qualitatively in the same way, but exhibits larger variations since [FORMULA].

[FIGURE] Fig. 11. Stellar luminosity (top) and mass loss rate (middle) as a function of time during a thermal pulse on the upper AGB according to stellar evolution calculations with mass loss by Blöcker (1995). Data are taken from the same evolutionary track as before, but refer to the thermal pulse occurring before the one shown in Fig. 7. The lower panel shows the gas (solid) and dust (dotted) velocity as a function of time at a radial distance of [FORMULA] cm as obtained from two-component radiation hydrodynamics calculations (DEXCEL) based on the stellar evolution sequence shown above. The thick line (middle and bottom panel) corresponds to the original data, the thin line (-317.005 yrs [FORMULA] -315.983 yrs) indicates the modified data used as input for the NEBEL simulations shown in Fig. 12. Times labels `1' to `11' serve as reference marks.

The concomitant variation of the gas and dust outflow velocity at [FORMULA] cm as calculated by DEXCEL is plotted in the bottom panel of Fig. 11. As stellar luminosity and mass loss rate decline, the driving of the outflow by radiation pressure on dust becomes progressively less efficient and the velocities reach a minimum at [FORMULA] km s-1, [FORMULA] km s-1, where the wind is assumed to be supported by shock wave pressure (cf. SSS98). It might even be that the gas falls back onto the star under these circumstances, but this situation cannot yet be handled by our code. In any case, this phase lasts for only about 1 000 yrs. It ends when luminosity and mass loss jump up, leading to an efficient acceleration of the dusty outflow. The gas velocity reaches a peak value of 14 km s-1, which is higher than the value before the onset of the flash (about 10 km s-1). This period of relatively high velocity is rather short, however. After another 1 000 yrs, the gas velocity is again down to 3 km s-1, this time for several 10 000 years, as [FORMULA] and [FORMULA] fall to vary low values, and gas and dust decouple completely.

This sequence of events may be viewed as a mass loss `eruption' which triggers a `storm', i.e a short phase of high velocity wind. The sharp velocity difference, in turn, inevitably leads to two-wind interaction and to the formation of a compressed gas shell. Note that the shell does not consist entirely of the matter ejected during the `eruption', but also of material that was ejected earlier, being swept up in the wind interaction region. Hence, the amount of mass accumulated in the shell is not directly related to the amount of mass involved in the mass loss event. Roughly, the gas at the center of the shell was expelled at the time of peak mass loss.

The formation of a compressed gas shell as a consequence of the helium-shell flash was not noticed in the original DEXCEL run documented in SSS98, because the resulting shell was of rather low amplitude and was washed out quickly by numerical diffusion. Only when we applied the NEBEL code to this problem, as usual with the hydrodynamical quantities provided by DEXCEL at [FORMULA] cm as input, the propagation of a compressed shell through the entire circumstellar envelope was clearly seen, although the density enhancement [FORMULA] was only moderate and the width of the shell increased considerably with distance, reaching [FORMULA] at [FORMULA] cm.

In order to further explore the potential of this scenario, we decided to slightly modify the velocity during the `eruption' phase: in the time interval -317.005 yrs [FORMULA] -315.983 yrs, we increased the outflow velocity applied as an inner boundary condition to NEBEL by a factor of 1.5. Leaving the gas density unchanged, this implies an enhancement of the mass loss rate by the same factor during this time span. The modified input data are shown by the thin lines in the lower panels of Fig. 11. We feel that a factor of 50% can easily be accommodated within the uncertainties of the model calculations, e.g. concerning the mass loss prescription and the dust grain properties.

The NEBEL results for the modified helium-shell flash scenario are illustrated in Fig. 12. The simulation clearly demonstrates the efficiency of the helium-shell flash in creating a large-amplitude, stable density structure traveling to the outer regions of the circumstellar envelope. The mass of the shell increases on its way out as a consequence of two-wind interaction.

[FIGURE] Fig. 12a-d. In the same way as before, this figure illustrates the creation and evolution of a shell of enhanced gas density as a consequence of the (modified) helium-shell flash shown in Fig. 11. In each of the upper three panels, there are 11 different curves corresponding to times labeled `1' to `11' in Fig. 11. These results were obtained with the NEBEL one-component hydrodynamics code using [FORMULA], [FORMULA], [FORMULA] from the DEXCEL two-component radiation hydrodynamics calculations at [FORMULA] cm (shown partly in the lower panel of Fig. 11) as an inner boundary condition. Artifacts beyond [FORMULA] cm should be ignored.

The NEBEL simulations suggest that the relative shell width is essentially constant with radial distance: [FORMULA] (lower panel of Fig. 12). Note that this number is in close agreement with the observed relative width of the CO shell around TT Cyg (Olofsson et al. 1998, 2000).

The average propagation velocity of the compressed shell is of the order of 14 km s-1, while, according to Olofsson et al. (1998, 2000), the expansion velocity of TT Cyg's CO shell is 12.6 km s-1, a remarkable coincidence. A closer inspection of the data reveals that the propagation of the shell is slightly decelerated, decreasing from [FORMULA] km s-1 at [FORMULA] cm to [FORMULA] km s-1 at [FORMULA] cm. We attribute this deceleration to the loss of momentum as a consequence of sweeping up `slow' gas at the outer edge of the shell at decreasing support from the `fast' wind at the inner edge of the shell. It is important to note that we again find a velocity plateau at the center of the shell (see also upper panel of Fig. 13), which is a typical feature of supersonic wind interaction (double-shock structure). This is consistent with the observational fact that the velocity dispersion indicated by the line width of CO emission from TT Cyg's shell is very low, of order of only 1 to 2 km s-1 (Olofsson et al. 1996, 2000).

[FIGURE] Fig. 13. Radial distribution of gas velocity (top), particle number density (middle) and gas temperature / pressure (bottom, solid / dashed) at time [FORMULA] yrs (between points 5 and 6), as obtained with the NEBEL one-component hydrodynamics code for the modified helium-shell flash scenario based on the stellar evolution sequence with mass loss shown in Fig. 11. Symbols (+) indicate the numerical grid. At this time the parameters of the AGB star are [FORMULA] K, [FORMULA], and for the gas shell we measure [FORMULA], [FORMULA].

The amplitude of the quantity [FORMULA] is decreasing somewhat with distance, indicating that the mass of the shell grows in a non-linear way with radial distance. At [FORMULA] cm, we measure a shell mass (integral over the range [FORMULA] [FORMULA] FWHM) of [FORMULA], a peak density of [FORMULA] g cm-3, and a density enhancement relative to the `background' of [FORMULA], while [FORMULA], [FORMULA] g cm-3, [FORMULA] at [FORMULA] cm. Again, it is worth mentioning that these numbers are in general agreement with the estimate of the mass of the CO shell around TT Cyg and the gas density density within the shell as given by Olofsson et al. (1998, 2000): [FORMULA] and [FORMULA] g cm-3.

Shock heating of the shell material seems to be negligible in the presence of molecular radiative cooling, as may be seen from the undisturbed temperature profile across the shell shown in the lower panel of Fig. 13. The kinetic temperature of the compressed shell equals the dust temperature, [FORMULA] K at [FORMULA] cm. This is consistent with the finding by Olofsson (2000), [FORMULA] K, in particular if we assume the material to be clumpy (see below). Since the temperature is uniform across the shell, pressure and density gradients always have the same sign, and hence there is no reason for the shell to become Rayleigh-Taylor unstable.

We have performed similar calculations for the subsequent helium-shell flash (depicted in Fig. 7). Here the temporal variations of luminosity and mass loss rate are qualitatively similar, but occur on a somewhat higher mean level; the peak mass loss rate is almost a factor of 2 higher. In consequence, the outflow velocity is no longer a monotonic function of the mass loss rate during the `eruption' as in the previous helium-shell flash (see Fig. 11). Rather, the outflow velocity decreases sharply at the time of maximum [FORMULA], because the radiation pressure on dust is momentarily reduced due to the high optical depth of the dust shell produced by the heavy mass loss. Under these circumstances, the rapid velocity variations during the flash-induced mass loss `eruption' eventually lead to the formation of a double shell . We have not yet studied this more complex case in detail. We just point out that, evidently, the circumstellar signature of a helium-shell flash depends sensitively on the amplitude of [FORMULA](t) and [FORMULA](t) during the flash, which in turn depends on the stellar core mass and the evolutionary stage on the AGB.

In summary, the results of these simulations covering the helium-shell flash and the ensuing phase of rapidly decreasing mass loss, clearly demonstrate that the sharp peak in luminosity and mass loss rate associated with the occurrence of a helium-shell flash triggers a short period of strongly enhanced wind speed. This, in turn, must lead to two-wind interaction which has the potential to produce very narrow circumstellar shells of compressed gas ([FORMULA], [FORMULA]). Note that it is not necessary to invoke unknown physical processes for this scenario to operate. All that is needed is some plausible fine-tuning.

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Online publication: May 3, 2000
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