A&A 384, 452-459 (2002)
DOI: 10.1051/0004-6361:20020022

An improved mass-loss description for dust-driven superwinds and tip-AGB evolution models

A. Wachter 1,2 - K.-P. Schröder 1 - J. M. Winters 3 - T. U. Arndt 2 - E. Sedlmayr 2


1 - Astronomy Centre, CPES, University of Sussex, Falmer, Brighton BN1 9QJ, UK
2 - Technische Universität Berlin, Zentrum für Astronomie und Astrophysik, PN 8-1, Hardenbergstr. 36, 10623 Berlin, Germany
3 - Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany

Received 7 November 2001 / Accepted 4 January 2002

Abstract
We derive an improved description of dust-driven stellar mass-loss for the cool winds of carbon-rich tip-AGB stars. We use pulsating wind models in which the mass loss is driven by radiation pressure on dust grains, for C-rich chemistry. From a larger set of these models, selected for representative dynamical (pulsational velocity amplitude $\Delta v$, period P) and chemical (the $\epsilon_{\rm C}/\epsilon_{\rm O}$ abundance ratio) input parameters, an improved approximative mass-loss formula has been derived which depends only on the stellar parameters (effective temperature $T_{\rm eff}$, luminosity L and mass M). Due to the detailed consideration of the chemistry and the physics of the dust nucleation and growth processes, there is a particularly strong dependence of the mass-loss rate $\dot{M}$ (in $M_{\odot}/$yr) on $T_{\rm eff}$: $\log{\dot{M}} = 8.86 - 1.95 \cdot \log{M/{M_{\odot}}} - 6.81 \cdot \log{T/{\rm K}} + 2.47 \cdot \log{L/{L_{\odot}}}$. The dependence of the model mass-loss on the pulsational period has explicitly been accounted for in connection with the luminosity dependence, by applying an observed period-luminosity relation for C-rich Miras. We also apply the improved mass-loss description to our evolution models, and we revisit their tip-AGB mass-loss histories and the total masses lost, in comparison to our earlier work with a preliminary mass-loss description. While there is virtually no difference for the models in the lower mass range of consideration ( $M_{\rm i} = 1.0$ to $\approx 1.3 \, M_{\odot}$), we now find more realistic, larger superwind mass-loss rates for larger stellar masses: i.e., $\dot{M}$ between $\approx$0.4 and $1.0 \times 10^{-4}$ $M_{\odot }$/yr for $M_{\rm i}$ between 1.85 and 2.65 $M_{\odot }$, removing between 0.6 and 1.2 $M_{\odot }$, respectively, during the final 30000 yrs on the tip-AGB.

Key words: stars: carbon - stars: circumstellar matter - stars: evolution - stars: interiors - stars: late-type - stars: mass loss


1 Introduction

Strong stellar mass-loss by dust-driven "superwinds'' (Renzini 1981) on the tip-AGB has long been identified as a major mechanism to recycle stellar material and to drive the chemical evolution of the galaxy. This applies to stars which do not undergo a supernova explosion ( $M_* \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil $\displaystyl... ...ineskip\halign{\hfil$\scriptscriptstyle ...) but, instead, lose several tenths to over one solar mass in a strong "cool'' stellar wind during their final $\approx$$ 30\,000$ years (Peimbert 1981; Kwok 1981) to form a planetary nebula (PN). During this brief phase of its evolution, the star becomes a very cool, strongly dust-enshrouded, long-period variable (LPV) object, and it exhibits mass-loss rates of the order of 10-4 $M_{\odot }$/yr, as derived from observations and models of LPVs (e.g., Knapp & Morris 1985; Winters et al. 2000). Because of the interpretational difficulties encountered with LPVs, there has long been a considerable interest in a theoretical description of tip-AGB mass-loss from both, an atmospheric and an evolutionary point of view.

Early descriptions of dust-driven mass-loss used for evolution models (Bowen 1988; Vassiliadis & Wood 1993; Blöcker 1995) did not take into account the highly temperature dependent chemistry of the dust-formation process, but considered the main energy input to be related to the pulsations of LPVs, rather then being dependent on the interaction of the radiation field with the freshly formed dust grains in the wind acceleration region. For this reason, the derived dependence of the mass-loss rate on the actual stellar parameters, especially on the effective temperature $T_{\rm eff}$, was in need of considerable improvement. Nevertheless, the magnitude of the total mass lost on the tip-AGB by those models did already match the evolutionary context, i.e., the final-mass/initial-mass relation as derived by Weidemann (1987, 1997), as that was just a matter of adopting the right scaling of the mass-loss description (Groenewegen et al. 1995).

A much more sophisticated approach to dust-driven mass-loss has been developed over the past 10 years, in terms of pulsating, hydrodynamic and self-consistent numerical models, which describe in detail the complex physics of a dust-driven wind and the dust-formation process around a cool AGB star (Fleischer et al. 1992; Sedlmayr 1994; Sedlmayr & Winters 1997), for a given set of stellar parameters and carbon-rich chemistry. Mass-loss rates are then obtained by averaging the outflow of the time-dependent models.

Since individual wind models are very computer time intensive, a first attempt was made to describe the model mass-loss in terms of the stellar and other input parameters by Arndt et al. (1997), using a multi-dimensional, maximum-likelihood fitting procedure on the set of C-rich model mass-loss rates available at that time. By averaging over the existing range of non-stellar (dynamical and chemical) input parameters, a mass-loss description was obtained which depends on the principle stellar parameters, $T_{\rm eff}$, L and M, only. That description differs remarkably from other approaches in that it is very sensitive to the effective temperature of the star. In fact, the characteristics of the macroscopic system is strongly influenced by the temperature-sensitivity of the microscopic dust-formation process.

In using the mass-loss description derived by Arndt et al. (1997) with tip-AGB evolution models, Schröder et al. (1999) found good agreement with observed properties of "superwinds'', and even with some observed short bursts of strong mass-loss, suggesting that the gross picture was about right. Meanwhile, however, there has been progress on several points which has created a need for revising the mass-loss description of Arndt et al. (1997):

With consideration of the above points, we here provide an improved mass-loss formula for dust-driven, C-rich winds. We then apply it to tip-AGB evolution models to investigate the significance of the new formula to superwind properties and total mass-loss yields by comparison with our previous work (Schröder et al. 1999; Schröder & Sedlmayr 2001).

2 Deriving the improved mass-loss formula

Our description of the mass loss during the tip-AGB is based on an substantially updated set of self-consistent, dynamical wind models for dust-forming, carbon-rich atmospheres. The range of model parameters covered matches the observed range and is given by Table 1. These models include time-dependent hydrodynamics, (equilibrium) chemistry, (stationary, grey) radiative transfer and carbon dust formation, growth and evaporation processes. A recent description of the wind model code has been given by Winters et al. (2000).


 

 
Table 1: Range of the parameters covered by the selected model grid.
M [$M_{\odot }$] 0.8-1.2
T [K] 2200-3000
L [$L_{\odot}$] 3500-15000
P [d] 104-1000


The hydrodynamical wind models used to calculate the mass loss rates treat in detail the circumstellar environment around pulsating long-period variable stars and therefore require the prescription not only of the three fundamental stellar parameters (i.e. stellar mass M, effective temperature $T_{\rm eff}$, luminosity L) plus the abundance ratio $\epsilon_{\rm C}/\epsilon_{\rm O}$ of carbon over oxygen. In addition, two parameters are required for the prescription of the inner boundary condition which simulates a pulsating stellar surface as a means of mechanical energy input. This inner boundary condition is provided by a sinusoidal variation of the innermost gridpoint in the hydro-model, parameterized by a velocity amplitude $\Delta v$ and a pulsation period P ("piston approximation'').

From the output of the time-dependent models, a mass-loss rate is obtained by averaging the outflow between r = 30 and 60 R*, and over a time span of 20 pulsation periods. So far, the models consider only carbon-rich winds, i.e. $\epsilon_{\rm C} / \epsilon_{\rm O} > 1$. This ratio determines the amount of carbon available for dust formation, making the reasonable approximation that all oxygen and an equal amount of carbon is locked up in CO molecules. To obtain a physically relevant representation of the theoretical mass-loss rates given by the large set of computed wind models, we applied the multidimensional maximum-likelihood method (as described by Arndt et al. 1997). The input parameters for each wind model are completely independent of each other. Hence, the data set to be described consists of a maximum of 6 independent (M, $T_{\rm eff}$, L, $\epsilon_{\rm C}/\epsilon_{\rm O}$, Pand $\Delta v$) and one dependent variable ($\dot{M}$) for each model. The mass-loss formula should be preferably simple and is therefore assumed to be linear: $\log \dot M_{\rm fit} = a_0 + a_1 \cdot \log x_1 + \cdots + a_n \cdot \log x_n$.

In order to also give a mass-loss approximation described only by the three fundamental stellar parameters M, L and $T_{\rm eff}$, Arndt et al. (1997) simply averaged over the mass-loss rates from models with otherwise differing input parameters. Consequently, their simplified formula, which we used for our evolution models in earlier work (Schröder et al. 1999; Schröder & Sedlmayr 2001), neglects any possible dependence of the mass-loss on $\epsilon_{\rm C}/\epsilon_{\rm O}$, P or $\Delta v$.

To explore the dependence on any of these parameters and its significance, we first fitted the full set of models available up to date for all 6 parameters to obtain the power of each parameter in the updated representation. We then used the same multidimensional maximum-likelihood method on appropriately reduced parameter sets to compare the quality of the fits, i.e., taking the correlation coefficient of each mass-loss rate approximation $\log \dot M_{\rm fit}$ and the set of the model mass-loss rates $\log \dot M$. If, by dropping a certain parameter, the correlation coefficient decreased significantly, and if a parameter enters the mass-loss formula with a comparatively large power, its significance would need further investigation. In this process, we have given special consideration to the following points:

With the consideration of the above points, a set of 58 wind models (given in Table 2) has been selected. For applying the maximum-likelihood method on this final set, we now consider only 4 remaining independent input parameters, i.e., effective temperature $T_{\rm eff}$, luminosity L, actual mass M and pulsation period P.

For convenience, these parameters are normalized to physically representative reference values, i.e., 1 $M_{\odot }$ for the mass, 2600 K for $T_{\rm eff}$, 104 $L_{\odot}$ for luminosity, and 650 d for the period.


 

 
Table 3: Listing of the 58 wind models, selected by the criteria explained in the text, to realistically represent dust-driven, carbon-rich mass-loss of real tip-AGB objects.
No. M*/$M_{\odot }$ $T_{\rm eff}$/K $L_\star$/$L_{\odot}$ P/d $\langle \dot M \rangle$ $\langle \alpha \rangle_{\rm t}$ W-No. $\epsilon_{\rm C}/\epsilon_{\rm O}$
1 0.63 3000 8000 820 3.0e-5 2.50 w155 1.30
2 0.63 3500 8000 460 5.8e-6 2.76 w172 1.30
3 0.63 3500 8000 820 1.4e-5 5.30 w159 1.30
4 0.70 3000 12000 1100 6.2e-5 6.61 w156 1.30
5 0.70 3500 12000 650 1.0e-5 4.15 w174 1.30
6 0.70 3700 12000 650 6.0e-6 1.59 w176 1.30
7 0.80 2200 15000 300 1.1e-4 5.42 w38 1.30
8 0.80 2400 7500 104 1.4e-5 3.94 w125/2 1.50
9 0.80 2550 7500 104 6.0e-6 3.16 w130/1 1.50
10 0.80 2600 3500 400 4.9e-6 1.30 w135 1.30
11 0.80 2600 4000 400 6.5e-6 1.38 w46 1.30
12 0.80 2600 5000 300 7.4e-6 1.34 w49/14 1.30
13 0.80 2600 5000 350 6.4e-6 1.74 w110 1.30
14 0.80 2600 5000 400 1.3e-5 1.82 w44 1.30
15 0.80 2600 5000 500 1.3e-5 1.88 w111 1.30
16 0.80 2600 5000 600 1.6e-5 1.96 w112 1.30
17 0.80 2600 6000 400 1.6e-5 2.28 w51 1.30
18 0.80 2600 7000 450 1.9e-5 2.61 w60 1.30
19 0.80 2600 7500 300 1.0e-5 2.38 w113 1.30
20 0.80 2600 7500 450 2.5e-5 2.61 w48 1.30
21 0.80 2600 7500 600 5.1e-5 1.96 w114 1.30
22 0.80 2600 7500 800 3.6e-5 2.28 w141 1.30
23 0.80 2600 10000 640 5.0e-5 2.02 w63 1.30
24 0.80 2600 12000 800 7.0e-5 4.65 w61 1.30
25 0.80 2600 15000 1000 9.9e-5 3.84 w62 1.30
26 0.80 2700 5000 300 3.9e-6 1.47 w167/2 1.30
27 0.80 2700 5000 350 5.6e-6 1.76 w168/4 1.30
28 0.80 2800 5000 400 6.1e-6 1.26 w52 1.30
29 0.80 3000 6000 400 3.8e-6 1.85 w47 1.30
30 0.80 3000 7500 400 9.0e-6 2.78 w124/2 1.50
31 0.80 3000 7500 450 8.3e-6 8.49 w32 1.80
32 0.80 3000 7500 650 1.0e-5 4.86 w31 1.80
33 0.80 3000 15000 300 1.7e-5 3.70 w30 1.50
34 0.80 3000 15000 650 3.0e-5 8.90 w28 1.50
35 0.80 3000 15000 800 4.1e-5 9.32 w29 1.50
36 0.84 3000 20000 1200 7.9e-5 10.07 w157 1.30
37 0.84 3500 20000 880 2.0e-5 5.47 w177 1.30
38 0.84 3700 20000 710 1.1e-5 4.07 w178 1.30
39 0.94 3000 25000 1300 8.8e-5 11.30 w158 1.30
40 0.94 3500 25000 1000 2.3e-5 6.77 w180 1.30
41 0.94 3700 25000 810 1.2e-5 2.85 w181 1.30
42 0.94 3900 25000 1300 2.5e-5 11.90 w183 1.30
43 1.00 2400 12000 300 2.6e-5 3.15 w122/1 1.30
44 1.00 2400 12000 500 5.9e-5 4.04 w144 1.30
45 1.00 2400 12000 600 7.7e-5 4.09 w120 1.30
46 1.00 2400 12000 800 7.9e-5 3.71 w145 1.30
47 1.00 2600 10000 640 4.3e-5 2.64 w64 1.30
48 1.00 2600 10000 650 2.6e-5 8.78 w13 1.80
49 1.00 2800 6000 400 4.6e-6 1.41 w169/5 1.35
50 1.00 2800 7000 400 5.4e-6 1.21 w78 1.30
51 1.00 2800 8000 400 5.1e-6 1.68 w97/1 1.30
52 1.00 2800 10000 640 1.8e-5 2.92 w69 1.30
53 1.00 2900 10000 578 1.3e-5 3.09 w71 1.30
54 1.00 2900 10000 578 9.6e-6 1.98 w72 1.25
55 1.20 2600 7000 400 6.1e-6 1.05 w53 1.30
56 1.20 2800 10000 400 1.3e-5 2.90 w109 1.80
57 1.20 2800 10000 400 1.6e-5 3.37 w108/6 1.50
58 1.20 2800 10000 400 9.8e-6 2.06 w126 1.40


Based on that final set of models, the following mass-loss description is obtained:

                           $\displaystyle \log \dot{M}_{{\rm fit}}$ = $\displaystyle -4.52 - 6.81 \cdot \log (T_{{\rm eff}} / 2600 \, \rm K)$  
    $\displaystyle + 1.54 \cdot \log (L / 10^4 \, L_{\odot}) - 1.95 \cdot \log (M / M_{\odot})$  
    $\displaystyle + 0.959 \cdot \log (P / 650 \, \rm d).$ (1)

It shows a very good correlation (with a coefficient of 0.965) with the set of model mass-loss rates. The typical deviation of $\dot M_{\rm fit}$ from any of the actual rates represented by it, is $\pm$20%. This is demonstrated by Figs. 1a-1c where the logarithmic ratio of the mass loss rates is plotted against M, $T_{\rm eff}$ and L respectively.


     \begin{figure} \par\psfrag{M / Msun}{\scriptsize$M$\space / $M_{\odot}$ } \psfr... ...ver luminosity]{ \includegraphics[width=8.8cm,clip]{h3281f3.eps} } \end{figure} Figure 1: Deviation of the mass-loss rate given by the fit formula from the model mass-loss rate represented by it. The dashed lines mark a deviation of 20%.
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In order to substitute the dependence on P, we now apply the above mentioned period-luminosity relation. Transformed to our reference values, it is virtually identical with $\log (P/650 \, {\rm d}) = 0.965 \cdot \log (L/10^4 \, L_{\odot})$. We can therefore substitute the period term by an additional power of 0.93 of the luminosity. This finally reduces the approximative mass-loss formula to depend on the three fundamental stellar parameters M, L, and $T_{\rm eff}$, only, but without neglecting significant effects from a dependence of the mass-loss on any other parameters (P, in particular):

                           $\displaystyle \log \dot{M}_{{\rm fit}}$ = $\displaystyle -4.52 - 6.81 \cdot \log (T_{{\rm eff}} / 2600 \, \rm K)$  
    $\displaystyle + 2.47 \cdot \log (L / 10^4 \, L_{\odot}) - 1.95 \cdot \log (M / M_{\odot})$ (2)

where $\dot M_{\rm fit}$ is given in units of $M_{\odot }$/yr.

3 Application to tip-AGB evolution and revised mass-loss yields

In order to demonstrate the relevance and significance of the improved mass-loss formula for dust-driven, C-rich winds, we applied it to our evolution models in the exact same way as described by Schröder et al. (1999) and Schröder & Sedlmayr (2001). In particular, we use the same description of convection and overshooting, the critical luminosity for the onset of dust-driven winds, and of any prior mass-loss.


  \begin{figure} \par\includegraphics[angle=270,width=\columnwidth]{h3281f4.eps} \end{figure} Figure 2: Tip-AGB and final 105 yrs of mass-loss history for a C-rich evolution model with $M_{\rm i} = 1.10$ $M_{\odot }$ and the improved mass-loss description - note the short burst of mass-loss as already discussed by Schröder et al. (1998, 1999). The label denotes the actual mass (in $M_{\odot }$) at that time.
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  \begin{figure} \par\includegraphics[angle=270,width=\columnwidth]{h3281f5.eps} \end{figure} Figure 3: As in Fig. 2, but for $M_{\rm i} = 1.30$ $M_{\odot }$, with the improved mass-loss description. This is the minimum initial mass required to obtain a regular superwind.
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By comparison, the resulting tip-AGB evolution models and their mass-loss histories are virtually the same in the lower range of initial stellar masses considered here (i.e., $M_{\rm i} = 1.0$ to $\approx$1.3 $M_{\odot }$, see Figs. 2, 3). A proper, continuous superwind sets in only for initial masses of $\approx$1.3 $M_{\odot }$ and larger (as found before), while short bursts of strong mass-loss are obtained for less massive tip-AGB models. Our new model with $M_{\rm i} = 1.1$ $M_{\odot }$ produces the same short ( $\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil $\displaystyle ...800 yrs) burst of mass-loss (nearly 10-5 $M_{\odot }$/yr), with the same properties as found by us before (Schröder et al. 1999), and removes obout 0.007 $M_{\odot }$ in that event. This is in excellent agreement with the latest observations of stunningly thin, spherical CO shells, especially the one of TT Cyg observed by Olofsson et al. (2000), who derive a total shell mass of 0.007 $M_{\odot }$. In addition, their observed outflow velocity (12.6 kms-1) of the shell is in very good agreement with the outflow velocity of our respective wind model.

Towards larger initial masses (see Figs. 4, 5), we now obtain higher superwind peak mass-loss rates and larger fractions of stellar mass lost by the superwind altogether: The $M_{\rm i} = 1.85$ $M_{\odot }$ model yields $4 \times 10^{-5}$ $M_{\odot }$/yr and a total loss of 0.62 $M_{\odot }$ during the final 30000 yrs (before: 0.50 $M_{\odot }$). The $M_{\rm i} = 2.65$ $M_{\odot }$ model even develops a superwind mass-loss peak of about $1 \times 10^{-4}$ $M_{\odot }$/yr, removing 1.22 $M_{\odot }$ (before: 0.80 $M_{\odot }$) in that phase.

The new results agree very well with the characteristics of PNs, and we believe that they now match even better than before. Basically, the new mass-loss description yields larger rates for stars with slightly higher $T_{\rm eff}$, M and for larger L. These stellar parameters are the ones reached by more massive models, which have their tip-AGB at higher luminosities and slightly larger $T_{\rm eff}$. This means, more massive stars produce denser circumstellar shells and, eventually, more massive PNs.


  \begin{figure} \par\includegraphics[angle=270,width=\columnwidth]{h3281f6.eps} \end{figure} Figure 4: Tip-AGB and final 105 yrs of mass-loss history for a C-rich evolution model with $M_{\rm i} = 1.85$ $M_{\odot }$ and the improved mass-loss description. Otherwise as Fig. 2.
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  \begin{figure} \par\includegraphics[angle=270,width=\columnwidth]{h3281f7.eps} \end{figure} Figure 5: As in Fig. 4, but for $M_{\rm i} = 2.65$ $M_{\odot }$. Note the larger peak mass-loss rate for larger initial stellar mass.
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In total, we re-computed 27 C-rich evolution models and their mass-loss histories, for initial stellar masses in the range of 1.0 to 2.8 $M_{\odot }$. The basic characteristics of these models are listed in Table 4. This table compares exactly to Table 2 of Schröder & Sedlmayr (2001). The now stronger mass-loss rates of more massive models lead to significantly larger yields during the superwind phase, which now exceed the AGB mass-losses prior to the superwind, giving more weight to the C-rich superwind yields for their contribution to the galactic mass re-injection rate. The same can be seen from Fig. 6, which compares to Fig. 8 of Schröder & Sedlmayr (2001), and which shows the total mass lost during each evolution phase relevant to stellar mass-loss, i.e., the Red Gant Branch (RGB), the Asymptotic Giant Branch (AGB, excl. superwind) and the superwind (final 30000yrs).

For the same reason, slightly smaller final masses are now obtained, approximately $M_{\rm f}/{M}_{\odot} = 0.55 \cdot (M_{\rm i}/{M}_{\odot})^{0.27}$, which is in even better agreement with the observed initial-mass, final-mass relation of Weidemann (1997) than before: for $M_{\rm i} = 2.5$ $M_{\odot }$, Weidemann (1997) derived $M_{\rm f} = 0.68$ $M_{\odot }$, while we obtain 0.70 $M_{\odot }$ (instead of 0.72 $M_{\odot }$ by Schröder & Sedlmayr 2001).

Looking at a complete stellar sample of stars, distributed in the range of $M_{\rm i} = 1.0$ $M_{\odot }$ to 2.5 $M_{\odot }$ with a logarithmic IMF representative of the solar neighbourhood stars ($\propto$ $ M_{\rm i}^{-1.8}$, see Schröder & Sedlmayr 2001), we find a re-injection of 57% of the total stellar mass involved, which is the same fraction as derived before by us (Schröder & Sedlmayr 2001). Again, however, the difference lies in the mass fraction lost during the C-rich superwind, which we now find to reach nearly 22% (was 17%), while the fraction lost during the prior AGB phases is nearly 29% (was 33%). Again, this enhances the importance of the C-rich winds on the tip-AGB relative to the cool but still O-rich winds prior to dust-driven mass-loss, in terms of their contribution to the galactic mass re-injection.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{h3281f8.eps} \end{figure} Figure 6: Mass lost by stars of different initial stellar mass (x-axis) during the RGB (front), the AGB (middle, excl. superwind) and the superwind (back, final 30000yrs) phases.
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Table 4: Listing of the revised evolution models, giving the initial stellar mass $M_{\rm i}$, the masses lost during the RGB, AGB (excl. superwind) and the superwind (final 30000yrs) phases, and the final mass $M_{\rm f}$ of the remnant WD.
$M_{\rm i}$ $\int{\dot{M}_{\rm RGB}}$ $\int{\dot{M}_{\rm AGB}}$ $\int{\dot{M}_{\rm SW}}$ $M_{\rm f}$  
1.00 0.24 0.20 -- 0.55  
1.05 0.16 0.30 -- 0.56  
1.10 0.12 0.38 0.01 0.56 1
1.15 0.11 0.41 0.03 0.57 1
1.20 0.09 0.47 0.03 0.58 1
1.25 0.08 0.40 0.15 0.59 1
1.30 0.08 0.30 0.28 0.60  
1.35 0.07 0.35 0.28 0.60  
1.40 0.07 0.37 0.31 0.61  
1.45 0.06 0.39 0.34 0.61  
1.50 0.06 0.39 0.38 0.62  
1.55 0.06 0.40 0.42 0.62  
1.60 0.05 0.41 0.46 0.63 2
1.65 0.04 0.42 0.51 0.63  
1.70 0.04 0.42 0.55 0.63  
1.75 0.03 0.43 0.60 0.64  
1.80 0.03 0.45 0.62 0.64  
1.85 0.02 0.50 0.63 0.64  
1.90 0.02 0.50 0.68 0.65  
1.95 0.01 0.55 0.69 0.65 3
2.05 0.001 0.58 0.79 0.66  
2.15 -- 0.59 0.87 0.67  
2.25 -- 0.63 0.92 0.68  
2.35 -- 0.64 1.00 0.69  
2.50 -- 0.67 1.11 0.70  
2.65 -- 0.70 1.22 0.71  
2.80 -- 0.74 1.32 0.72  
Sample: 6.3% 28.7% 21.7% 43.3% 4
1 Only brief superwind burst(s).
2 Onset of core overshooting on MS at $M_{\rm i} \approx 1.6$ $M_{\odot }$.
3 RGB evolution ends with He flash for $M_{\rm i} \le 1.95$ $M_{\odot }$.
4 Fractions of mass lost by a stellar sample (1.0 to 2.5 $M_{\odot }$, IMF $\propto M_{*}^{-1.8}$).


4 Discussion

Despite significant improvement, there are a few points which will require some more work in the future. In particular, we believe that the most critical of the remaining simplifications concerns the description of the mechanical energy input, $\Delta v$ in particular, for which a fixed value may not be applicable to the whole range of tip-AGB models. Hopefully, a larger set of models, in combination with a better knowledge of stellar parameters of individual pulsating objects, will reveal any kind of relationship between the best choice of $\Delta v$ with any fundamental stellar parameter, i.e., with L. Then, any $\Delta v$ term in the mass-loss description could be considered in a similar way as we treated the P term.

We would like to emphasize that the here presented mass-loss formula applies only to carbon-rich mass-loss, and for strong dust-driven winds. The chemistry of dust-formation in oxygen-rich winds is substantially different and requires its own modelling. Such work is on the way (Jeong et al. 1999) and a specific oxygen-rich mass-loss description is in preparation.

The application of the improved mass-loss approximation to the computation of individual evolution models, as well as to collective mass-loss yields, demonstrates the far-reaching implications of any improvement to stellar and galactic astrophysics.

Despite some remaining issues, we may conclude that we have derived a significantly improved and physically meaningful mass-loss description for carbon-rich, dust-driven winds of tip-AGB objects, based on a large set of theoretical wind models, selected to represent observed objects best. The application of the new formula to evolution models yields more massive superwinds and PNs towards the larger stellar masses as derived before with a preliminary mass-loss representation, and the results are in good agreement with observed properties of PNs.

Acknowledgements

This work has been supported by a visitor grant held by A. Liddle, given to JMW for a visit of the Sussex Astronomy Centre, and by a Marie Curie Fellowship of the European Community programme HUMAN POTENTIAL under the contract number HPMT-CT-2000-00096 given to AW to work at the Physics & Astronomy department of Sussex University. KPS wishes to acknowledge conference travel support, granted to him by the Royal Society.

References

 

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