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

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5. Comparison with observations

It is presently a widely accepted idea that single carbon stars are formed during the evolution on the AGB when C-rich material is dredged-up from the interior to the O-rich photosphere just after a helium shell flash (Iben & Renzini 1983, Willems & de Jong 1988). The transition from O- to C-rich chemistry could be very fast, taking place on a time scale of the order of years (envelope mixing time). The model proposed by Willems and de Jong assumes that mass loss is strongly reduced, compared to the previous phase of evolution, when the photosphere becomes C-rich for the first time. In consequence, a detached O-rich shell expands away from the visible C-rich star. Up to now 19 carbon rich stars with oxygen-rich envelopes are known (see e.g. Kwok et al. 1997).

Simple modeling (without hydrodynamical effects and with constant mass loss rate) of the O-rich envelopes moving away from the star, performed by Willems & de Jong (1988) and by Chan & Kwok (1988), accounts not only for C stars with silicate dust emission, but also for the optically visible C stars with 60 µm excess originating from dust that is too cold to show any signature in the mid-infrared wavelength range. Excess at 60 and/or 100 µm is usually interpreted as a signature of a detached shell. In a few cases, re-analysis of IRAS data, observations with ISO or even high-resolution ground-based observations have confirmed the existence of such shells (Waters et al. 1994; Izumiura et al. 1996, 1997; Olofsson et al. 1996).

There is a well known problem with the scenario proposed in the late eighties. Namely, it seems that the detached envelopes around visible carbon stars are rather carbon-rich than oxygen-rich (see e.g. Zuckerman 1993). The estimated life times of C stars is between 105 and 106 years (e.g. Claussen et al. 1987), while the life time of a detached shell is about 104 years (cf. Figs. 11, 12). Consequently, the fraction of carbon stars with detached shells should be significantly lower than 10% if a mass loss interruption occurs only once at the beginning of the carbon star's life.

According to the mass loss formula applied in this work, interruption of mass loss happens a few times during the live of a carbon star (assumed here to last at least 350 000 years). Our approach explains in a natural way the observational fact that most of the envelopes around C stars with 60 µm excess are carbon rich. Note, however, that we did not consider the transition phase when both sorts of dust (silicate in the outer and carbon in the inner part of the envelope) could co-exist.

5.1. The distribution of AGB stars in the IRAS two-color diagrams

Over the time interval of 350 000 yrs covered by our simulations (Fig. 2), the computed IRAS colors [FORMULA], [FORMULA], [FORMULA] vary in correspondence with the dust composition and its distribution inside the shell heated by the central source of radiation. Each of the four thermal pulses produces a distinct loop in the IRAS two-color-diagrams (lower panel of Figs. 14 to 17). The evolutionary behavior of the computed colors can be compared with the distribution of the observational data in the same color-color diagrams (upper panel of Figs. 14 to 17).

[FIGURE] Fig. 14. Top : Distribution of observed carbon stars in the IRAS (60/25 versus 25/12) two-color-diagram. The data for `optical' C stars (classified as C-rich according to their optical spectra; grey squares), `classical' C stars (showing SiC emission at [FORMULA] 11.3 µm; black circles), and `extreme' C stars (thick dust shells, central stars not detectable in visual surveys; asterisks) are courtesy of K. Volk (priv. comm.) Only objects with fluxes of quality 3 at 12, 25, 60, and 100 µm are plotted. Naturally, there is an overlap between the `optical' and `classical' C stars. Bottom : Computed evolution of the standard carbon star model in the same IRAS two-color-diagram, based on the stellar evolution sequence shown in Fig. 2. The hydrodynamical simulation covers four thermal pulses during the final 350 000 years of AGB evolution, each of which produces a corresponding loop in the two-color-diagram, distinguished here by different symbols: squares (B-C), diamonds (D-E), plus-signs (F-G), and asterisks (H-I), plotted at equidistant time intervals of 1000 years, while small triangles are used otherwise (A-B, C-D, E-F, G-H, I-J). The sequence starts somewhere in region VII, reaches the position marked "t=0" at the end of the AGB and leaves the diagram to the right when entering the post-AGB phase (the zigzag motion after [FORMULA] is of numerical origin).

[FIGURE] Fig. 15. Same as Fig. 14, but presenting the distribution of observed and computed carbon stars in the IRAS (100/60 versus 25/12) two-color-diagram to include also the 100 µm flux. The big "star" near position (-0.6,-0.45) in the lower diagram represents the black body point for [FORMULA] K.

[FIGURE] Fig. 16. Top : Distribution of observed oxygen stars in the IRAS (60/25 versus 25/12) two-color-diagram. The data for oxygen stars with silicate emission (black circles) and with silicate absorption (asterisks) are courtesy of K. Volk (priv. comm.); while objects with 60 µm excess (grey squares) are from Zijlstra et al. (1992). Only objects with fluxes of quality 3 at 12, 25, 60, and 100 µm are plotted. Bottom : Computed evolution of the standard oxygen star model in the same IRAS two-color-diagram, based on the stellar evolution sequence shown in Fig. 2. As for the carbon star sequence (Fig. 14) the four thermal pulses during the final 350 000 years of AGB evolution lead to four distinct loops in the two-color-diagram (for an explanation of the different symbols see caption of Fig. 14). Again, the time interval between two adjacent symbols is 1000 years. The sequence starts off somewhere in region IIIa, and leaves the diagram to the right at the end of the AGB evolution.

[FIGURE] Fig. 17. Same as Fig. 16, but presenting the distribution of observed and computed oxygen stars in the IRAS (100/60 versus 25/12) two-color-diagram. Again, the big "star" near position (-0.6,-0.45) in the lower diagram represents the black body point for [FORMULA] K.

From the comparison presented in Figs. 14-17 it is obvious that the problematic co-existence of the both C-based and Si-based dust, as required by Ivezi & Elitzur (1995) in the framework of steady state models, is absolutely not necessary if the time-dependence due to evolutionary changes of the stellar parameters and mass loss rate are taken into account. Our tracks not only cover the areas with observational data but also explain (roughly) the observed relative population densities of the different "cells" in the IRAS two-color-diagrams. Since the theoretical points are plotted at equidistant time intervals (of 1000 years), their number density is a direct measure of the probability of finding objects in different parts of the diagram. It should be kept in mind, however, that our models are based on a single evolutionary track. Only when the model calculations finally cover a representative sample of initial stellar masses, it will be possible to constrain the variety of proposed mass loss laws from a detailed comparison with observations.

5.1.1. Comparison for carbon stars

In Figs 14 and 15 we compare our theoretical results for the carbon-rich envelope with corresponding observations in two different IRAS color-color diagrams. In Fig 14 the regions introduced by van der Veen & Habing (1988) are shown for reference. The agreement with observational data is quite remarkable in both diagrams.

What is the most important point, Blöcker's mass loss formula can explain carbon stars with 60 µm excess: The objects located in region VIa (which cannot be understood in the framework of steady state models) are carbon stars just having suffered a thermal pulse and are presently in the middle of the subsequent phase of "mass loss interruption" (cf. Willems & de Jong, 1988; Chan & Kwok, 1988) with mass loss rates reduced by at least an order of magnitude.

The only C stars which cannot be readily explained by our models are the extreme C stars (marked by asterisks in Figs. 14 and 15). As has been suggested by Volk et al. (1992), those stars are probably near the end of the AGB evolution and are characterized by huge mass loss. However, our models with amorphous carbon cannot explain their positions in the IRAS two-color diagrams even for mass loss rates as high as 10[FORMULA].

We suggest that the extreme C stars could be explained by the presence of graphite grains, which have a very broad feature around 30 µm (Draine & Lee, 1984, see also Fig. 1 of Paper I). This feature, contributing to the 25 µm IRAS band, would shift our tracks to redder colors. Graphite grains would also explain a slight disagreement in the [FORMULA] color seen in Fig. 15, because the extinction cross section of graphite decreases more steeply with wavelengths than for amorphous carbon, so this color would become slightly bluer, resulting in a better agreement with the observational data. Possibly, the structure of the circumstellar dust grains changes from amorphous to more ordered forms at the highest mass loss rates.

5.1.2. Comparison for oxygen stars

In Figs. 16 and 17, a similar comparison has been performed between oxygen-rich stars and models based on dust composed of "astronomical" silicates. It is now established from observations that mass loss is subject to interruptions also in the case of O-rich stars (Zijlstra et al. 1992). As a natural consequence of such mass loss variability, detached shells are expected to develop repeatedly (not only once during the transition from oxygen-rich to carbon-rich surface composition). Agreement between our track and the observational data in Fig. 16 is fairly good, with the exception of the oxygen-rich stars with silicate absorption (marked by asterisks). However, as has been already shown by Bedijn (1987), O-rich stars in the upper part of regions IIIb and IV require different dust properties to be explained. Namely, for wavelengths longer than about 30 µm, the absorption coefficient should decrease less steeply with wavelength than it does in the case of "astronomical" silicates.

Again stars with 60 µm excess are pretty well explained by mass loss reduction associated with the decline of luminosity after the helium shell flash. Note, that the predicted time scales agree quite well with the observed density of data points.

Some of the observational points in region VIb might actually be more massive O-rich supergiants moving off the giant region with reduced moss loss. They should be removed from this sample of AGB stars once identified.

In the [FORMULA] versus [FORMULA] diagram (Fig. 17) the situation is somewhat worse. While stars with detached shells are still quite well explained by our models, there is a lot of data points in the upper part of diagram. These objects showing silicate emission (probably) cannot all be due to contamination with massive O-rich supergiants.

To understand the situation for these stars with silicate emission, we should point out that O-rich stars are somewhat less evolved than carbon stars. They are therefore somewhat less luminous and have lower mass loss rates, implying smaller 100 µm fluxes on average. In consequence, contamination from the cirrus emission could be more severe in this case, resulting in potentially spurious 100 µm fluxes.

Again, changes in the optical properties of silicates as proposed by Bedijn (1987) would better explain the observed positions of stars with silicate absorption ([FORMULA] will increase).

5.1.3. The influence of fcond

As is to be expected, the photometric properties derived from our dynamical models depend somewhat on the assumed dust condensation efficiency [FORMULA]. A test calculation for the carbon star sequence with [FORMULA], but otherwise identical parameters, revealed that the loop corresponding to the `first' thermal pulse (cf. lower panel of Fig. 14), remains almost unchanged, while the amplitude of the `second' and `third' loop is reduced by about a factor of 2. The last loop is again identical. The latter coincidence is easily explained because, for the last loop, [FORMULA] was also valid for the original sequence and we have already seen that the choice of [FORMULA] has no significant influence on the dynamics of the outflow. The close agreement of the `first' loop is explained by the fact that during the "mass loss interruption" after the `first' thermal pulse, the density of hot dust in the inner shell is not sufficient to produce significant thermal emission: the flux at 12 and 25 µm is totally dominated by the stellar radiation. A further reduction of the dust density ([FORMULA]) does not make any difference. Of course, the "old" dust seen at [FORMULA] 60 µm is also identical since it was produced at times of high mass loss when [FORMULA]. Together, this implies that one may expect extended loops to correspond to the previous thermal pulses (not covered by our simulation). For the subsequent thermal pulses, however, the minimum mass loss rate lies at increasingly higher levels (cf. Fig. 2) and the dust emission at 12 and 25 µm is no longer negligible. This explains why the shape and extent of the `second' and `third' loop depends on the choice of [FORMULA].

5.2. The spectral energy distribution of S Scuti

Observationally, several objects with excess dust emission around 60 to 100 µm are known to have well-detached shells (e.g. Olofsson et al. 1996). A prominent example is the well-studied carbon star S Scuti (cf. Groenewegen & de Jong 1994). We selected one post-flash hydrodynamical model from our standard carbon star sequence which comes closest to the observed spectral energy distribution of S Scuti at 60 and 100 µm (see top panel of Fig. 18). Obviously, the model is somewhat too hot to fit the observed fluxes at UV, visual and near infrared wavelengths. At this instant, the model's central star has an effective temperature of [FORMULA] K.

[FIGURE] Fig. 18. Top : Observed spectral energy distribution of the well-known carbon star S Scuti (different symbols; data from Groenewegen & de Jong, 1994) compared with two different theoretical spectra. The first one (solid grey, model SS1) is taken from the model sequence shown in Fig. 11 at time [FORMULA] yrs (close to [FORMULA]) such that the model spectrum fits the fluxes at 60 and 100 µm. The corresponding dashed line shows the assumed input spectrum of the AGB star at the center of the detached dust shell, a blackbody with [FORMULA], [FORMULA]. The second spectrum (dot-dashed, model SS2) is taken from a similar sequence which was computed with [FORMULA] reduced by 25% relative to the original track. The optimum fit is obtained at time [FORMULA] yrs, when [FORMULA] K, [FORMULA]. The remarkable agreement with the observed spectral energy istribution is a natural result of the temporal variation of the mass loss rate seen in Fig. 10. Remaining differences in the UV may disappear when using a realistic stellar flux distribution instead of assuming a black body spectrum (corresponding dashed line). Bottom : Emergent intensity at [FORMULA] 12, 25, 60, and 100 µm as a function of impact parameter for model SS2. Intensity is in arbitrary units, normalized to be identical for all wavelengths at [FORMULA] cm. A detached dust shell is clearly visible at all wavelengths.

In order to obtain a better overall fit, we have computed another sequence with [FORMULA] reduced by 25% relative to the original track, keeping the stellar luminosity unchanged (the stellar radius increases accordingly) as well as retaining the dust properties and numerical parameters. Somewhat surprisingly, we found the dynamics of the circumstellar matter to be considerably affected: the outflow velocity is markedly reduced and the density correspondingly enhanced. Obviously, the efficiency of the radiative dust acceleration is reduced significantly because the cooler central star radiates fewer photons at shorter wavelengths where the dust absorption cross section is largest. Although the wind velocity is now unrealistically low, the enforced time variation of the mass loss rate Fig. 10 again leads to the development of detached dust shells and corresponding excess emission at 60 and 100 µm.

For this sequence, the observed spectral energy distribution is matched most closely at time [FORMULA] yrs, when [FORMULA] K, [FORMULA] (model SS2, dot-dashed line in the top panel of Fig. 18). The agreement is quite remarkable even though we have not fine-tuned the model to fit this particular object. Clearly, the model could have been improved by adjusting the stellar parameters, the dust-to-gas ratio and the dust size distribution. Remaining differences in the UV will probably disappear when using a realistic stellar flux distribution instead of assuming a black body spectrum. This is, however, not the purpose of the present work.

For completeness, we show in the bottom panel of Fig. 18 the emergent intensity at different wavelengths ([FORMULA] 12, 25, 60, and 100 µm) as a function of the impact parameter for model SS2. We note that a detached dust shell is clearly visible at all wavelengths, unlike the situation shown in the top panel of Fig. 13, but very similar to the model of the oxygen-rich dust shell displayed in the bottom panel of Fig. 13. Obviously, the outflow velocities in the wind of the cool carbon star are comparable to those in the oxygen star with the original (higher) [FORMULA].

5.3. The detached dust shell of Y CVn

In the previous section we have seen that, around the time of minimum mass loss rate, the computed radial intensity distribution shows a local maximum at distances of a few [FORMULA] cm from the central star, corresponding to a ring-like structure in the surface brightness (lower panel of Fig. 18). The computed intensity maps show the ring-like structure to be most prominent in the far infrared, [FORMULA]m. Indeed, such detached dust shells have been detected by IRAS (Waters et al. 1994; Izumiura et al. 1997) and by ISO (Izumiura et al. 1996).

In an earlier publication (Steffen & Szczerba 1997), we have compared the radial intensity distribution at [FORMULA] 100 µm computed from hydrodynamical model SS2 (see Sect. 5.2) with the surface brightness map of the carbon star Y CVn obtained at [FORMULA] 90 µm with the ISOPHOT camera on board the Infrared Space Observatory (ISO) by Izumiura et al. (1996). The comparison with the synthetic data shows a remarkable qualitative agreement (for details see Fig. 6 of Steffen & Szczerba 1997). In both cases the signature of a detached dust shell is clearly discernible as a local maximum in the brightness distribution. In the model the shell's emission peaks near [FORMULA] cm where the dust temperature is [FORMULA] K.

5.4. Transition to the post-AGB phase: IRAS 17437+5003

Observationally, the end of the AGB evolution is characterized by an optically thin dust shell with a somewhat hotter stellar remnant shining through. Obviously, the mass loss rate must have dropped by orders of magnitude on a very short time scale. The physical reasons for this rapid decline of mass loss, however, are yet unknown. In the empirical mass loss modeling by Blöcker (1995) the rate is coupled to the period of the fundamental radial pulsational mode, P, forcing the transition from the high AGB rate to the much lower Reimers rate (Reimers 1975) to occur between periods of P = 100 and 50 days. It happens that this procedure leads to a mass loss reduction of about two orders of magnitude within 100 years (cf. Fig. 2 and top panel of Fig. 19). Note that the zero point of our time scale is defined such that [FORMULA] d.

[FIGURE] Fig. 19. Top : Mass loss rate adopted during the transition from the AGB towards higher effective temperatures (blow-up of rightmost part of Fig. 2, using a non-linear time-axis to resolve details near t=0). Middle : Radial distribution of the dust density at 3 selected times indicated in the upper panel ([FORMULA]) as obtained from the sequence computed with dust grains composed of "astronomical" silicates. The outer parts of the dust density profile are considerably steeper than suggested by the [FORMULA] density law indicated by the dotted reference line. Bottom : Corresponding emergent spectral energy distributions (solid lines) at times [FORMULA]. The dashed lines indicate the corresponding intrinsic spectra of the central star, for which the effective temperatures are given in the legend.

As mentioned in Sect. 4.1.3, we have extended our hydrodynamical simulations several hundred years into the post-AGB regime, using the mass loss law shown in the upper panel of Fig. 19. Indeed we find a rapid detachment and thinning of the dust shell as the density of the newly formed hot dust decreases sharply at the end of the AGB evolution (due to the largely reduced mass loss rate and the sharply increasing dust drift velocity) and gives no detectable signature. The time evolution of the dust density in the inner parts of the shell during the transition to the post-AGB phase is shown in the middle panel of Fig. 19. The rapid depletion of the inner dust shell, caused by the sudden drop of the mass loss rate near [FORMULA], is clearly seen. Note also that the outer parts of the actual dust density profile are considerably steeper than suggested by the [FORMULA] density law indicated by the dotted reference line, a consequence of the steadily increasing mass loss rate near the tip of the AGB. The mass loss minimum after the final helium shell flash, centered at [FORMULA] yrs (top panel of Fig. 19) gives rise to a deep local minimum of the gas density located near [FORMULA] cm at the beginning of the post-AGB evolution (middle panel of Fig. 19).

The corresponding changes of the emergent spectral energy distribution are illustrated in the bottom panel of Fig. 19 for a simulation based on silicate dust grains. During the covered time interval of about 570 yrs, the strong silicate absorption feature near 10 µm is seen to disappear rapidly with advancing shell detachment, while the thermal emission of the dust becomes progressively more concentrated at far infrared wavelengths. At the same time, the previously totally enshrouded AGB remnant becomes visible, giving rise to a characteristic double-peaked energy distribution. The synthetic spectrum computed for time [FORMULA] yrs shows a stunning agreement with the observed spectral energy distribution of the well-known post-AGB object IRAS 17436 +5003 = HD 171 796. The comparison is presented in Fig. 20. In principle, the very good agreement of observed and computed spectral energy distribution indicates that the observed relation between the evolution time scale of the star and the detachment time scale of its circumstellar envelope is basically matched by theory. This implies that the mass loss formula adopted for the end of the AGB evolution and the wind velocities predicted by our model closely reflect the actual situation.

[FIGURE] Fig. 20. Spectral energy distribution at time [FORMULA] (solid, also shown in the bottom panel of Fig. 19) compared with observed fluxes of IRAS  17436+5003 = HD 161 796 (diamonds; data from Hrivnak et al. 1989). Note that the observations seem to indicate this object to be slightly hotter ([FORMULA] [FORMULA] 7000 K) than the central star of the model ([FORMULA] [FORMULA] 6300 K) which, however, was not designed to fit this particular post-AGB star!

Finally, in the top panel of Fig. 21, we compare the radial distribution of the gas density obtained from the sequences with amorphous carbon dust and with dust of "astronomical" silicates, respectively, at the beginning of the post-AGB phase (about 50 and 150 yrs after the end of the AGB evolution). The outward motion of the inner edge of the gas shell, caused by the sudden drop of the mass loss rate near [FORMULA], over the considered time interval of 100 yrs is clearly seen. The differences between the carbon-based and the silicate-based models in the inner part of the shell are caused the the different outflow velocities. We have mentioned before (Sect. 4.2) that the radiation pressure on silicate grains is less efficient than for grains of amorphous carbon and hence the outflow velocities are lower for oxygen-rich dust shells. This can also be seen in the lower panel of Fig. 21. Over the main parts of the shell, however, the density structure is very similar for both type of models. As was seen for the dust density before (middle panel of Fig. 19), the actual gas density also drops considerably faster with radial distance than [FORMULA], reflecting the mass loss history during the previous 50 000 years of evolution.

[FIGURE] Fig. 21. Top : Radial distribution of the gas density about 50 (left-) and 150 yrs (right-shifted curves) after the end of the AGB evolution as obtained from the sequences with amorphous carbon dust (solid) and with dust of "astronomical" silicates (dashed), respectively. The outward motion of the inner edge of the gas shell, caused by the sudden drop of the mass loss rate near [FORMULA], over the considered time interval of 100 yrs is clearly seen. Note that the actual gas density drops considerably faster with radial distance than suggested by the [FORMULA] density law indicated by the dotted reference line. Bottom : Similar comparison for the gas velocities.

The density and velocity structure shown in Fig. 21 constitute the starting point for the subsequent development of a planetary nebula. The data are ideally suited as initial conditions for the hydrodynamical modeling of planetary nebulae and have already been used successfully for this purpose (Schönberner et al. 1997, 1998). The results will be the subject of a more detailed discussion in a forthcoming paper.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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