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Astron. Astrophys. 337, 149-177 (1998)

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4. Results of the numerical simulations

4.1. Time evolution of a circumstellar shell with amorphous carbon dust - a paradigmatic overview

The example results presented in the following are based on the stellar evolution sequence described in the previous section. The data shown in Fig. 2 are used as a time-dependent inner boundary condition for the hydrodynamical simulation of the long-term evolution of the circumstellar gas/dust shell as explained in Sect. 2.5. The dust is assumed to condense as grains of amorphous carbon with properties as given in Table 1. The initial velocity at the dust formation point [FORMULA] is assumed to be constant ([FORMULA] km/s) throughout the sequence.

4.1.1. From the initial model through the `first' thermal pulse

The initial model was obtained as a steady state solution (see Paper I) for stellar parameters and mass loss rate at time [FORMULA] yr (corresponding to time "A" indicated in Fig. 2). Subsequently, the steady state solution was modified in the outer regions such that [FORMULA] g/cm3 and the outflow velocity is zero to represent the presence of the interstellar medium (cf. Sect. 2.3). Obviously, it is unrealistic to introduce the interstellar medium in this way since it will have been pushed much farther out by the stellar outflow during the previous evolution. The only reason for doing so is to demonstrate that the interaction of the slow AGB wind with the ISM is not producing any major observable features in the stellar spectrum; in particular, it will be seen that it has nothing to do with the formation of detached shells.

Velocity field, density structure, and spectral energy distribution of the model serving as the initial condition for the time dependent calculations are shown in the left-hand panels of Fig. 3. The properties of steady state models have been described in detail in Paper I. We notice that typically such models show a constant gas outflow velocity of the order of 10 km/s over a wide radial range, leading to a radial density dependence [FORMULA]. The ratio of dust and gas density is approximately constant and corresponds to the assumed dust-to-gas ratio of [FORMULA]=0.0015, multiplied by the factor [FORMULA] ([FORMULA] since [FORMULA], see Eqs.19 and 20).

[FIGURE] Fig. 3. Velocity field, density structure, and spectral energy distribution at two different times for the model sequence based on the stellar evolution data shown in Fig. 2. Dust is assumed to consist of grains of amorphous carbon . The left hand panels show the initial conditions for the time dependent calculations, corresponding to a steady state solution for stellar parameters and mass loss rate at time [FORMULA] yr (time "A" in Fig. 2), modified in the outer regions such that [FORMULA] g/cm3 and the outflow velocity is zero to represent the presence of the interstellar medium. The right hand panels are for time [FORMULA] yr, roughly corresponding to the time of minimum mass loss rate between "B" and "C" (see Fig. 2). Upper panels : gas (solid) and dust velocity (dotted) as a function of radial distance. Middle panels : radial distribution of gas (solid) and dust density (dotted). In both panels, the two faint dashed lines indicate the [FORMULA] (gas-/dust) density laws representing the initial steady state solution. Lower panels : stellar input spectrum (dot-dashed) and the emergent spectral energy distribution (solid) which is the result of processing of the stellar radiation by the dusty envelope. Time and current stellar mass are given in the frame headers.

After about 18 000 yrs, the stellar luminosity and mass loss rate have dropped to a local minimum and conditions have changed significantly. The right hand panels of Fig. 3 show the situation at time [FORMULA] yr, roughly corresponding to the time of minimum mass loss rate between "B" and "C" (see Fig. 2). Due to the low mass loss rate the gas density in the inner part of the shell has decreased by almost two orders of magnitude. In addition, [FORMULA] is now only 0.01 since [FORMULA], as may be seen from Fig. 2. As a result, the dust density has dropped by almost four orders of magnitude (see middle panels) and the frictional coupling between gas and dust is no longer sufficient to drive the outflow, as indicated by the low gas outflow velocity, [FORMULA] 3 km/s, and the high velocity of the dust component, [FORMULA] 22 km/s. Rather the gas is driven solely by virtue of the "shock waves", [FORMULA] (see Eq. 2).

However, in the outer part of the shell ([FORMULA] cm), velocity and density still more or less correspond to the much higher mass loss rate before the onset of the thermal pulse. This means that a detached shell has been created which, in the present case, is much more pronounced for the dust component than for the gas component. The main reason for the different behavior of the dust and gas components is the [FORMULA]-dependent factor [FORMULA] and the fact that the dust velocity in the inner parts has increased by a factor of [FORMULA] while the gas velocity has decreased by a factor of [FORMULA]. In the emergent spectrum, the detached dust shell is clearly seen as a conspicuous excess emission beyond [FORMULA]m (lower panels).

Finally, we notice that at the boundary between the stellar wind and the "interstellar medium" some matter has been piled up, giving rise to a local density enhancement near [FORMULA] cm. At the same time, this interaction front has been pushed outward somewhat, while in the outermost parts of the model the interstellar medium is still unaffected (note that [FORMULA] in the outer parts of the model). In the considered wavelength range of the emergent spectrum we do not see any signature indicating the interaction of the stellar wind with the ISM.

4.1.2. Further evolution

During the further evolution the mass loss rate increases steadily for about 80 000 years and replenishes the inner low-density regions of the shell. After some time the dust condensation becomes efficient again ([FORMULA]) and the number density of dust grains exceeds the critical limit for driving the outflow. The left-hand panels of Fig. 4 show the situation at time [FORMULA] yr (time "D" in Figs. 2 and 10), just before the onset of the next thermal pulse. The velocity and density structure closely resemble that of the initial steady state model, except in the outermost regions that still exhibit the signature of the previous thermal pulse. The pronounced density minimum near [FORMULA] cm is the result of the extended mass loss rate minimum between "B" and "C". The emergent SED is not sensitive to the conditions in the outermost parts of the shell and hence is very similar to that of the initial steady state solution, again showing a monotonic decrease of the flux towards longer wavelengths. Since luminosity and mass loss rate are somewhat higher than initially, the gas outflow velocity is somewhat higher while the drift velocity [FORMULA] has slightly decreased. Due to the slightly increased overall density, the emergent spectral energy distribution has shifted slightly to the red. Notice that the "interstellar medium" has meanwhile almost been pushed out of the computational domain. Its presence is only indicated by the low velocities near the outer boundary.

[FIGURE] Fig. 4. Same as Fig. 3 but for times [FORMULA] yr (left hand panels), roughly corresponding to time "D" indicated in Fig. 2 and [FORMULA] yr (right hand panels), roughly corresponding to the time of minimum mass loss rate between "D" and "E" in Fig. 2. Notice that, in this and the following figures of this type, the [FORMULA] reference lines in the density plots are scaled to match the actual density profiles at [FORMULA] cm, unlike in Fig. 3 where they represent the initial steady state.

At the next minimum of the mass loss rate at time [FORMULA] yr (right-hand panels of Fig. 4), the coupling between gas and dust has again become insufficient to drive the outflow due to the low dust density, as indicated by the slow (3 km/s) gas flow which is supported only by the assumed "shock wave pressure". In the outer regions the gas velocity still is about 11 km/s, corresponding to the outflow velocity before the mass loss rate began to decrease. In the inner regions, the dust density is again very low and follows a [FORMULA] law out to [FORMULA] cm. Farther out, however, we see a distinct increase of the dust density with a local maximum near [FORMULA] cm, representing again a detached dust shell . In the emergent spectrum, this excess concentration of dust is seen as an unmistakable excess emission at [FORMULA]m.

The general hydrodynamical structure at this time is very similar to that during the mass loss minimum of the `first' thermal pulse. Note, however, that the detached dust shell is somewhat broader due to the fact that duration of the period of reduced dust condensation efficiency ([FORMULA]) is significantly shorter for the `second' thermal pulse, as may be seen by following the dotted horizontal lines in Fig. 2.

The same sequence of events occurs when we follow the evolution from the `second' to the `third' thermal pulse. The general hydrodynamical structure and emergent energy distribution before the onset of the `third' thermal pulse, shown in the left-hand panels of Fig. 5, is again very similar to the situation encountered before the previous thermal pulse. Due to the higher mass loss rate the overall density is somewhat larger, the drift velocity is somewhat reduced, and more of the stellar radiation at visible wavelengths is absorbed by the circumstellar dust and reradiated in the infrared. The interstellar medium has now been completely pushed out of the computational domain.

[FIGURE] Fig. 5. Same as Fig. 3 but for times [FORMULA] yr (left hand panels), roughly corresponding to time "F" indicated in Fig. 2 and [FORMULA] yr (right hand panels), roughly corresponding to the time of minimum mass loss rate between "F" and "G" in Fig. 2.

During the following minimum (right-hand panels of Fig. 5), the mass loss rate no longer falls below [FORMULA]. Hence, the dust condensation efficiency is hardly reduced ([FORMULA]). Nevertheless, we see that the dust density is still insufficient to couple efficiently to the gas component and to drive the outflow. The gas velocity is still only about 3 km/s in the inner shell, while it is about 12 km/s in the outer shell. As before, a detached shell has formed. However, it is apparent that the radial structure of the dust shell is now much more similar to that of the gas shell. This is because the temporal variation of the factor [FORMULA] is now largely reduced and because the drift velocity in the outer parts is relatively small. Since the dust density in the detached shell is now significantly higher than during the previous thermal pulse, the absolute fluxes near 60 and 100 µm are now distinctly enhanced. At visual wavelengths, however, the shell is optically thin and the central star freely visible.

During the following 76 000 years of evolution the mass loss rate increases by a factor of 100. Shortly before the `fourth' and final thermal pulse occurs, the model AGB star has become completely obscured by the surrounding dust shell: the optical depth in the visual ([FORMULA]m) is [FORMULA]. The plots for time [FORMULA] yr (left-hand panels of Fig. 6) show that the outflow velocity is slightly lower than before the `third' thermal pulse. The reason is that due to the larger optical depth (densities have increased by a factor of 10) fewer high-energy photons are now available to accelerate the dust grains. In the outer parts the density structure deviates markedly from the usual [FORMULA] law, reflecting the more pronounced (compared to the earlier cycles) temporal increase of the mass loss rate during the previous evolution.

[FIGURE] Fig. 6. Same as Fig. 3 but for times [FORMULA] yr (left hand panels), roughly corresponding to time "H" indicated in Fig. 2 and [FORMULA] yr (right hand panels), roughly corresponding to the time of minimum mass loss rate between "H" and "I" in Fig. 2.

At the minimum at time [FORMULA] yr (right-hand panels of Fig. 6) the dust density is still sufficient to drive the outflow without the support of "shock waves", although only marginally. As before, a detached shell has formed due to the steeply decreasing mass loss rate. For the reasons given above, the spatial structure of the gas and dust shell are now almost identical. At a mass loss rate of [FORMULA]we find the visual optical depth to be [FORMULA], so the optical visibility of the central AGB star is still somewhat reduced. The hump in the spectral energy distribution near [FORMULA] 60 and 100 µm clearly indicates the presence of a detached dust shell.

4.1.3. End of the AGB evolution

At the end of the AGB evolution, when mass loss rate is at its maximum ([FORMULA], [FORMULA] in our example), the visual optical depth of the dust shell is [FORMULA] and the outflow velocity in the inner shell is quite uniform, about 10 km/s. This is at the lower end of outflow velocities observed in dusty carbon stars (cf. Habing et al. 1994, Fig. 8), indicating that the luminosity of our stellar model and/or the adopted dust-to-gas ratio is somewhat low. Increasing [FORMULA] and [FORMULA] by a factor of 2 would lead to a more typical outflow velocity close to 20 km/s according to the scaling relations given by Habing et al. (1994).

After the mass loss rate has dropped to the Reimers rate ([FORMULA]) at time [FORMULA] (left-hand panels of Fig. 7), density in the innermost shell has diminished by almost three orders of magnitude and the central star begins to shine through the expanding shell again ([FORMULA]). Since the dust in the inner regions is now exposed to the unobscured stellar radiation, which moreover now originates from a star with greatly reduced mass and rapidly increasing effective temperature, it is strongly accelerated and reaches a maximum velocity of [FORMULA] km/s. Despite the low coupling, the gas component is initially also strongly accelerated up to more than 20 km/s. The radial density distribution clearly reflects the steadily increasing mass loss rate before the end of the AGB evolution in the outer parts of the shell and the subsequent sharp drop from the AGB to the Reimers mass loss rate in the inner parts, resulting in a characteristic shape distinctly different from the simple [FORMULA] relation.

[FIGURE] Fig. 7. Same as Fig. 3 but showing the situation at the end of the AGB evolution at times [FORMULA] yr (left hand panels; [FORMULA] K, maximum dust velocity w = 50 km/s), roughly corresponding to time "J" indicated in Fig. 2, and at [FORMULA] yrs into the post-AGB evolution (right hand panels; [FORMULA] = 8500 K, maximum dust velocity w = 120 km/s).

We extended our hydrodynamical simulations somewhat into the post-AGB regime, using the mass loss prescription shown in the upper panel of Fig. 19. After 660 yrs of post-AGB evolution (see right-hand panels of Fig. 7) the gas/dust shell has completely detached from the star which has increased its effective temperature to [FORMULA] K. Due to the low Reimers mass loss rate the gas density in the inner part of the shell is very low, and the dust moves outward with high speed ([FORMULA] km/s) since it is largely decoupled from the gas. As a result, the inner shell is now essentially devoid of hot dust, even though still [FORMULA]: the visual optical depth is [FORMULA], and the central star is again unobscured at optical wavelengths. The dust density distribution during this phase of evolution produces a characteristic double-peaked spectral energy distribution shown in the lower right panel.

The transition from the AGB to the post-AGB phase is discussed in more detail in Sect. 5.4.

4.1.4. Summarizing the dynamical evolution of the shell

The sequence of events described in the previous sections demonstrates that the time-dependence (on stellar evolution time scales) of the density structure and velocity field of a circumstellar AGB shell is probably quite complex. In order to summarize the main features, we display in Fig. 8 the temporal evolution of the gas and dust velocity (top), the gas and dust density (middle) as well as the dust temperature (bottom) at an intermediate radial position, [FORMULA] cm, over the time interval of [FORMULA] yrs covered by the simulation discussed above.

[FIGURE] Fig. 8. Top : Gas velocity (solid) and dust velocity (dotted) as a function of time at a distance of [FORMULA]cm from the star (data re-interpolated from the moving grid to constant r). Data are taken from the model sequence computed with amorphous carbon dust, for which details are shown in Figs. 3 to 7. Middle : Similar plot of the gas density (solid) and the dust density (dotted). For a more direct comparison, the dust density has been divided by the dust-to-gas ratio, [FORMULA]. Bottom : Temporal variation of the dust temperature at [FORMULA] (upper) and [FORMULA]cm (lower curve). The general time-dependence of the dust temperature, including the sharp final increase reflects the variation of the stellar luminosity and effective temperature, modified by the temporal variation of the optical depth.

Obviously, the gas velocity prefers one of two distinct states: when the mass loss rate is below some critical limit, the coupling between gas and dust is insufficient to drive the outflow via radiation pressure on dust and the flow is essentially driven by the artificial acceleration term [FORMULA] representing the pressure of the shock waves produced by the stellar pulsation. In this state, [FORMULA] km/s, and the drift velocity of the dust relative to the gas is largest during these periods. For sufficiently high mass loss rates, on the other hand, the driving mechanism based on the radiation pressure on dust is taking over and leads to [FORMULA] km/s. At the same time the dust drift is relatively small due to the close coupling of dust and gas. Note that the duration of the periods of low outflow velocity decreases in the course of the AGB evolution, because on average [FORMULA] increases with time and falls less deeply below the critical limit for each consecutive thermal pulse.

The temporal variation of the gas density closely follows the variations of the mass loss rate enforced at the inner boundary of the shell. However, it is modified by the time-dependence of the gas velocity, leading to a reduced amplitude of the overall gas density fluctuations (since u is small when [FORMULA] is low and vice versa). One important point to note here is that while the extended periods of low mass loss rate are clearly seen in the [FORMULA] data at [FORMULA] cm, the short-term variations of the mass loss rate (labeled `2',`3 'in Fig. 10) occurring before the eventual long-term decline of [FORMULA] are not detectable. Obviously, the associated density fluctuations are smeared out quickly by hydrodynamical effects and have largely disappeared before reaching the outer regions of the shell. We have checked that the small-scale density structure does not disappear due to numerical limitations (e.g. due to the deteriorating spatial resolution of the numerical grid with radial distance). It is a consequence of the rather complex physical behavior of the system, related to the fact that, due to optical depth effects, the outflow velocity is a non-monotonic function of the mass loss rate. As a consequence, the density peak related to mass loss "eruption" `3' runs into the density depression related to the preceding short mass loss reduction `2', resulting in a partial cancellation of the density fluctuations. The usual assumption that the outflow velocity is independent of mass loss rate, often made in semi-empirical models of time-dependent dusty outflows (recent examples: Groenewegen & de Jong 1994; Shu 1997; Shu & Jones 1997), is clearly not valid in this case (for details see Steffen & Schönberner 1998).

Surprisingly, however, we see strong short-term density enhancements (of a factor of 3 to 4) near the end of the phases of low mass loss rate (see middle panel near [FORMULA] yrs). These are caused by local compression of the gas where matter with higher velocity runs into regions of lower velocity and higher density: the density peaks coincide with the times of steep velocity increase, i.e they are related to the switch over from a slow, "shock-driven" to a faster, "dust-driven" wind. Note that the transition from the slow to the fast wind happens quite abruptly when the mass loss rate exceeds a critical limit [FORMULA]. It has nothing to do with the assumed time-dependent dust condensation efficiency [FORMULA], but is a general property of the dust-driven wind: below a critical wind density, the coupling between dust and gas in insufficient and the gas outflow cannot be maintained; for wind densities even slightly above this limit, however, the outflow is accelerated efficiently. This property inevitably leads to an almost bimodal dependence of the outflow velocity on the mass loss rate, and consequently to the formation of wind interaction regions . These manifest themselves as thin shells of enhanced gas density (and probably enhanced gas temperature as well). We propose that these features are related to the origin of the very thin molecular shells detected in CO emission around a number of carbon stars by Olofsson et al. (1998). This point is further discussed in a separate publication (Steffen & Schönberner 1998).

The dust density closely follows the gas density, except for some phase shift caused by the fact that the dust velocity is, at times, significantly higher than the gas velocity. And, of course, the dust density is reduced by an additional factor of up to 100 whenever [FORMULA] (cf. Sect. 2.5), an effect more pronounced during the earlier thermal pulses.

The dust temperature [FORMULA] at [FORMULA] cm varies between 55 and 35 K. At the outer boundary of our model ([FORMULA] cm), [FORMULA] is close to 20 K. Remembering that [FORMULA] is computed as the radiative equilibrium temperature of the dust grains, it is evident that, at a fixed distance from the star, [FORMULA] depends on the stellar luminosity and effective temperature, as well as on the column density of dust between the central star and the considered radial position. In this sense the temporal variation of [FORMULA] seen in the lower panel of Fig. 8 can be understood as the combined effect of the temporal variation of [FORMULA], [FORMULA], and [FORMULA] which determines the column density, i.e. optical depth, in a non-local way. One can clearly see that the dust temperature increases during the mass loss minima when the absorption of the stellar radiation by the intervening dust diminishes and the flux spectrum incident on the dust grains shifts to shorter wavelengths. The final sharp rise of [FORMULA] is caused by the greatly reduced optical depth of the inner dust shell, combined with a considerable increase of the stellar effective temperature at the beginning of the post-AGB phase.

Finally, we point out that all the time-dependent hydrodynamical features discussed above do not depend on the assumed time-dependence of the dust condensation efficiency factor [FORMULA] (see Eqs.19, 20). A test run with [FORMULA] shows very much the same temporal behavior as seen in Fig. 8.

4.2. Model calculations for a shell with silicate dust

The model sequence described above was based on the assumption that the dust condenses as grains of amorphous carbon (AC). We have computed another sequence assuming the dust to condense as grains of "astronomical" silicates (AS) with properties given in Table 1 (see Sect. 2.2). In all other respects the two sequences are identical. In particular, they are based on the same stellar evolution sequence [FORMULA], [FORMULA], [FORMULA] and cover the same period of time.

We do not discuss here the very details of the hydrodynamical evolution of the circumstellar shell with silicate dust, since this case is in principle very similar to the one presented for amorphous carbon dust. As an example, we only show the hydrodynamical structure and emergent spectral energy distribution before and after the final thermal pulse (Fig. 9). This plot is made for the same instants of time as Fig. 6 in order to allow a direct comparison.

[FIGURE] Fig. 9. Velocity field, density structure, and spectral energy distribution at two different times from the model sequence computed with dust grains composed of "astronomical" silicates . The times are [FORMULA] yr (left hand panels), roughly corresponding to time "H" indicated in Fig. 2, and [FORMULA] yr (right hand panels), roughly corresponding to the time of minimum mass loss rate between "H" and "I" in Fig. 2. Note that the silicate features are seen in absorption before the thermal pulse and change into emission during the subsequent period of low mass loss rate. For comparison with the carbon star sequence, see Fig. 6.

The main differences arise from the different absorption properties of "astronomical" silicates. Assuming a fixed grain size of [FORMULA]m, their extinction cross section per unit mass is at least a factor of 5 lower than for amorphous carbon grains over the wavelength range 0.2 to 7 µm, which is centered on the maximum of the stellar spectral energy distribution. However, due to the presence of the "silicate features", the absorption cross section of AS exhibits strong local maxima near [FORMULA] and [FORMULA]m, where the extinction cross section per unit mass exceeds that of AC by up to a factor of 5. In comparison to the case of AC, this non-monotonic wavelength-dependence of the extinction efficiency of AS gives rise to a qualitatively different response to changes of the spectral energy distribution of the incident radiation.

The main difference, however, is the lower overall extinction efficiency of AS between 0.2 to 7 µm. This leads to a lower average acceleration and outflow velocity (especially during the earlier thermal pulses), even though we adopted a higher dust-to-gas ratio for AS ([FORMULA], instead of [FORMULA] for AC). As a consequence, the features in the density structure are closer to the star at a given time (compare Figs. 6 and 9). Also, the average density of the outflow is somewhat higher, and the phase shift between the fluctuations of gas and dust density are more pronounced. Due to the higher overall dust density, there is hardly any excess emission at [FORMULA] 60 and 100 µm in the example shown in Fig. 9.

In the following sections we will present a detailed comparison of the photometric properties derived from the sequences computed with amorphous carbon dust and with "astronomical" silicate dust, respectively.

4.3. Variation of emergent spectrum and surface brightness distribution during a thermal pulse cycle

In order to illustrate in more detail how the emergent spectral energy distribution changes during a typical thermal pulse cycle, we have chosen the pulse centered on [FORMULA] yrs as an example. The time-dependence of the mass loss rate during this thermal pulse is fully resolved in Fig. 10.

[FIGURE] Fig. 10. Close-up of Fig. 2, showing the temporal variation of the mass loss rate of our standard stellar model ([FORMULA]) during the `second' thermal pulse cycle, about 200 000 yrs before the end of the AGB evolution. For mass loss rates higher than [FORMULA] (upper dotted horizontal line) the dust condensation fraction is assumed to be 100% of the value given in the last row of Table 1, dropping smoothly to 1% of this value for mass loss rates below [FORMULA] (lower dotted horizontal line). The times labeled "1" to "10" (and "m") serve as reference points for the data shown in the following figures. The separation between two adjacent diamonds corresponds to 1000 yrs.

4.3.1. Model fluxes for carbon stars

For the "carbon star" model sequence computed with amorphous carbon dust, which has been described in detail in the previous sections, we have evaluated the emergent spectral energy distribution and the spatial distribution of the surface brightness as a function of time.

As this model runs through the `second' thermal pulse cycle, the emergent spectral energy distribution changes with time as illustrated in the upper panels of Fig. 11. At time [FORMULA] the SED corresponds to that of a steady state model with [FORMULA] K, [FORMULA] and [FORMULA]. About 15 000 years later (time [FORMULA]), the SED has shifted considerably toward shorter wavelengths. Beyond [FORMULA] 25 µm, however, the flux diminishes at a slower rate, leading to the development of a relative excess of emission in the IRAS pass bands centered on 60 and 100 µm. As was demonstrated above (Sects. 4.1.1, 4.1.2), this is characteristic of the presence of a detached dust shell. As mass loss resumes and replenishes the inner shell with hot dust, the spectral energy distribution gradually reddens and the flux maximum near 100 µm disappears. At time [FORMULA], about 64 000 years after time [FORMULA], the mass loss rate has reached its former value and the spectra at times [FORMULA] and [FORMULA] are practically indistinguishable.

[FIGURE] Fig. 11. Time sequence of spectral energy distributions over the thermal pulse cycle shown in Fig. 10, assuming the dust grains to be composed of amorphous carbon with properties as given in Table 1. The emergent spectra for times [FORMULA] to [FORMULA] (see labels in Fig. 10) are presented in the upper panel , while those for times [FORMULA] to [FORMULA] are displayed in the middle panel . Note the pronounced excess emission at [FORMULA] 60 and 100 µm at times [FORMULA] and [FORMULA]. The lower panel shows the corresponding loop of this object in the IRAS two-color-diagram, with positions at reference times [FORMULA] to [FORMULA] indicated by + signs (the positions for [FORMULA] and [FORMULA] coincide). Open diamonds outline the time evolution in steps of [FORMULA] yrs. Note that the excursion to region "VIb" ([FORMULA]) is of very short duration. The "star" in region I marks the position of a black body with [FORMULA] K. Subdivision into regions after van der Veen & Habing (1988).

This temporal variation of the emergent spectral energy distribution translates into an extended loop in leftmost part of the IRAS two-color-diagram, as shown in the lower panel of Fig. 11. Note that around the time of minimum mass loss rate the object spends several thousand years in region "VIa", at positions in the IRAS two-color-diagram which are quite remote from the main color-color relation valid for steady state models for amorphous carbon dust (cf. Fig. 14 of Paper I). In contrast, the excursion into region "VIb" ([FORMULA]) takes less than 1000 yrs.

The time evolution of the radial distribution of the surface brightness at [FORMULA] 100 µm, i.e. the emergent intensity projected onto the plane of the sky, is shown in the top panel of Fig. 13. At time [FORMULA], in the middle of the extended period of mass loss "interruption", the intensity distribution shows a local maximum at a distance of about [FORMULA] cm from the central star, corresponding to a ring-like structure on a surface brightness map. During the further evolution the position of maximum brightness moves outward. However, since this ring-like feature is rather broad (width [FORMULA] 10 times separation from the star, see also middle right panel of Fig. 4) and the intensity contrast is very low, it will be hard to detect, unlike the associated excess emission in the SED, which is a prominent feature.

4.3.2. Model fluxes for oxygen stars

For the "oxygen star" model sequence computed with dust consisting of "astronomical silicates" (see Sect. 4.2), we have evaluated the emergent spectral energy distribution and the spatial distribution of the surface brightness in exactly the same manner as for the "carbon star" sequence.

The temporal variation of the emergent spectral energy distribution over the `second' thermal pulse cycle is presented in the upper panels of Fig. 12. At time [FORMULA] the SED corresponds to that of a steady state model with [FORMULA]. The silicate features near [FORMULA] 10 and 20 µm are seen in emission (with some self-absorption in the center of the 10 µm feature). At time [FORMULA] the silicate features are still prominent. But only 3000 yrs later ([FORMULA]) they have essentially disappeared, since the inner parts of the shell are now almost devoid of "hot" dust. At the same time the "cool" dust, now located in a detached shell, stands out as an excess emission in the IRAS 60 and 100 µm pass bands. As mass loss resumes and replenishes the inner shell with hot dust, the spectral energy distribution gradually reddens, the silicate emission features reappear and the signature of the detached dust shell in the far infrared vanishes. As in the case of the carbon star model, the spectra at times [FORMULA] and [FORMULA] are practically indistinguishable.

[FIGURE] Fig. 12. Time sequence of spectral energy distributions (top and middle) and corresponding loop in the the IRAS two-color-diagram (bottom) over thermal pulse cycle shown in Fig. 10, based on the same stellar evolution sequence and presented in the same way as the results shown in Fig. 11, but now assuming the dust to be composed of "astronomical" silicates . Again, the colors at times [FORMULA] and [FORMULA] are almost identical. Excess emission at [FORMULA] 60 and 100 µm is again clearly seen at times [FORMULA] and t6, while at the same time the silicate features at 9.7 and 18 µm are essentially absent. In contrast to the very short excursion to region "IIIb" ([FORMULA]), the extended loop into region "VIa" takes more than 10 000 years.

[FIGURE] Fig. 13. Time evolution of the radial intensity distribution at [FORMULA] 100 µm showing the formation and development of a detached dust shell for part of the carbon star sequence (top) and part of the oxygen star sequence (bottom). Times [FORMULA] to [FORMULA] are identical to those used in Figs. 11 and 12, respectively. In addition, the intensity distribution has also been evaluated at time [FORMULA], the time of the local minimum of the mass loss rate at [FORMULA] yrs before the end of the AGB evolution (see also Fig. 10). Although intensity is given in arbitrary units, the scale is constant for each panel, so the relative variation of the intensity distribution within the two time sequences is reproduced correctly. Because of the higher dust velocity in the carbon star model (top), the dust shell is much more extended and located at greater radial distances than for the oxygen star model (bottom). In consequence, a detached dust shell is much more clearly visible in surface brightness maps produced for the oxygen star model than for the carbon star model, although in both cases the amount of excess emission at 60 and 100 µm is very similar (cf. Figs. 11 and 12, time [FORMULA]).

The corresponding loop in the IRAS two-color-diagram is shown in the lower panel of Fig. 12. Note that this loop is more extended, both in horizontal and vertical direction, than the loop traced out by the carbon star model. The main reason is the different wavelength dependence of the silicate opacity: the starting point (time [FORMULA]) is located in a lower position in the IRAS two-color-diagram because the opacity gradient in the far infrared is steeper for "astronomical" silicates ([FORMULA] while [FORMULA]); the loop starts farther to the right because due to the presence of the silicate emission features the fluxes at 12 and 25 µm are initially almost equal. Since the silicate features temporarily disappear in the course of the thermal pulse, the ratio of the fluxes at 12 and 25 µm varies over a larger range than in the case of amorphous carbon dust. Consequently, the loop is more extended in the horizontal direction. The reason for the larger extent in the vertical direction is not so obvious. A comparison of the dust density distributions in the oxygen and carbon star outflows, respectively, reveals that due to the lower outflow velocities the density gradients are somewhat steeper in the former case. This leads to a larger amplitude of the color variations for the oxygen star model.

Note again that in the middle of the time interval of low mass loss rate, the loop of our oxygen star covers several different regions of the IRAS two-color-diagram, spending thousands of years at positions which are quite distinct from the main color-color relation valid for steady state models with silicate dust (cf. Fig. 12 of Paper I).

The time evolution of the surface brightness distribution at [FORMULA] 100 µm, is displayed in the bottom panel of Fig. 13. At times close to minimum mass loss rate the intensity distribution shows a local maximum at a distance of about [FORMULA] cm from the central star. It is much narrower and has a higher intensity contrast than in the case of amorphous carbon. Because our simulations produce lower velocities for the oxygen-rich outflows, the detached dust shell has a higher density and is confined to a smaller width, so it may be more easily detected as a a ring-like structure on a surface brightness map. During the further evolution the position of maximum brightness moves outward at a speed of about 6 km/s while the emission from the detached shell fades away.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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