7 Conclusions

We have developed a detailed radiative transfer code to model circumstellar molecular line emission. The code also solves for the energy balance equation of the gas. It is found that the mass loss rate determination for low mass loss rate objects depends crucially on a number of assumptions in the CSE model, except on the dust properties, since the CO molecules are radiatively excited. On the other hand, the mass loss rate estimate for a high mass loss rate object depends essentially only on the temperature structure, and hence the uncertain dust parameters. In addition, for such an object the CO emission saturates, and becomes less useful as a mass loss rate measure. We also find that different lines respond differently to changes in the various parameters. Therefore, a reliable mass loss rate determination requires, in addition to a detailed radiative transfer analysis, good observational constraints in the form of multi-transition observations and radial brightness distributions.

This model has been applied to CO radio line observations of a large sample of optically bright N-type and J-type carbon stars on the AGB (69 objects). The sample is reasonably complete out to about 1kpc, and all stars (41) within $\sim$500pc of the Sun have been detected in circumstellar CO line emission. The derived mass loss rates span almost four orders of magnitude, $\sim$5 10 $^{-9}\,M_{\odot}$ yr^-1 to $\sim$2 10 $^{-5}\,M_\odot$ yr^-1, over which the physical conditions of the CSEs vary considerably. The fact that the model can be succesfully applied over such a wide range of environments gives us confidence in the results, even though we are aware of the fact that some assumptions are poorly constrained. The large majority of the stars have mass loss rates in a narrow range centered at $\sim$3 10 $^{-7}\,M_{\odot}$yr^-1, and it appears that very few AGB carbon stars ( 5%) lose matter at a rate less than $\sim$5 10^-8$M_{\odot}$yr^-1.

We find that the mass loss rate and the gas expansion velocity are relatively well correlated, but the scatter is large enough that we may conclude that the mass loss mechanism is able to produce a wide range of mass loss rates for a given expansion velocity. The mass loss rate is also well correlated with the pulsational period of the star, correlated with the stellar luminosity, and there is a weak trend with the stellar effective temperature, in the sense that the cooler stars tend to have higher mass loss rates. Also the gas expansion velocity is positively correlated with the period and the luminosity. We conclude that the mass loss rate increases with increased regular pulsation and/or luminosity, and that the expansion velocity increases with mass loss rate (for low mass loss rates) and luminosity. The observed trends are all supporting the common concensus that these winds are driven by radiation pressure on dust grains, and that pulsation may play an important role. Somewhat surprising there appears to be no dependence on the stellar C/O-ratio.

Our standard CSE model, assuming a single smooth expanding wind produced by a continuous mass loss, fails to reproduce the observational data for about 10% of the sample stars. Most notable among these are five stars with detached CSEs, presumably formed during a period of highly increased mass loss, and low present mass loss rates. We have found indications that the present mass loss rate is significantly higher for the star with the youngest dCSE, and that the shell mass may increase with shell age. Since our sample is reasonably complete, we can estimate that the time scale between mass ejections is about 10⁵years, if it is a repeatable phenomenon. An association with He-shell flashes is favoured. These objects have been the targets of a number of extensive, observational and theoretical studies.

For some of our sample stars there exist enough observational constraints to determine a combined dust parameter from the kinetic temperature structure. This result can be interpreted as a dust-to-gas mass ratio which increases with mass loss rate, but changes in the dust properties may also play a role. We also find that this means that the gas kinetic temperature in a carbon star CSE depends only weakly on the mass loss rate.

The size of the CO envelope is an important parameter in the mass loss rate determination, at least for the low mass loss rate objects. We have used published radial CO( $J=1\rightarrow 0$) brightness distributions, in combination with the radiative transfer code, to show that the CO photodissociation calculation by Mamon et al. 1988 gives reasonably accurate results. In a few cases the observed radial brightness distributions are clearly different than the model results, suggesting deviations from a simple r^-2 density law, and hence time-variable mass loss.

We estimate that carbon stars, of the type studied here, return on the order of 0.05$M_{\odot}$yr^-1 of gas to the ISM making them marginally important for the gas cycle in galaxies. More extreme carbon stars may contribute an order of magnitude more. However, they are probably not important as regards the origin of carbon.

Acknowledgements

We are grateful to Dr. F. Kerschbaum for generously providing estimates of some of the input parameters to the CO modelling, to Dr. R. Liseau for useful comments, and to an anonymous referee for constructive criticism which lead to an improved paper. Financial support from the Swedish Natural Science Research Council (NFR) is gratefully acknowledged.