6 Discussion

6.1 The mass loss rate distribution

Figure 10 shows the distribution of the mass loss rates obtained from the radiative transfer model for all sources (also shown is the distribution for the stars within 500pc of the Sun), as well as divided into variability groups (note that objects with known detached shells have not been included in this analysis). Since the statistics of the SRa stars are poor we have chosen to group them together with the SRb stars. The mass loss rate distribution for all stars is sharply peaked around the median mass loss rate of 2.8 10 $^{-7}~M_{\odot}$yr^-1 (the median mass loss rate for the stars within 500 pc of the Sun is 1.6 10 $^{-7}~M_{\odot}$yr^-1). It is interesting to note that this is roughly the rate at which the core mass is expected to grow due to nuclear burning (Schönberner 1983). Mira variables generally have larger mass loss rates than other variability types, while irregular variables and semiregulars appear to have very similar mass loss rate characteristics.

Based on the mass loss rate distribution for the sample of stars within 500pc, which we believe to be close to complete and for which we have detected all sources, we are able to draw some general conclusions. The sharp decline at mass loss rates below $\sim$5 10 $^{-8}\,M_\odot$yr^-1 is very likely real. Netzer & Elitzur (1993) estimate that a mass loss rate in excess of $\sim$10 $^{-7}\,M_{\odot}$yr^-1 is required to get a dust-driven wind. The exact limit is, however, sensitive to the adopted stellar and dust parameters. The drastically decreasing number of high mass loss rate objects is also real, although our selection criterion bias against these objects, and can be explained by the fact that carbon stars of the type discussed here, i.e., mainly low mass ones (Claussen et al. 1987), only for a limited time, or possibly never, reach high mass loss rates.

Figure 11: The derived mass loss rate plotted against the gas expansion velocity of the CSE. Sources with known detached shells are indicated with triangles (open triangles represent dCSEs, while filled ones represent aCSEs). $\dot{M}$ scales as $v_{\mathrm e}^{\xi}$, with $\xi =2.5$. Also shown is the correlation coefficient r

6.2 Mass loss and envelope kinematics

The v_e-distribution for this sample of stars has already been shown and discussed by Olofsson et al. (1993a). However, a new comparison between the mass loss characteristics $\dot{M}$ and v_e is warranted considering the more reliable mass loss rates obtained in this paper, Fig. 11. We find a clear trend that v_e increases with $\dot{M}$, v_e$\propto$ $\dot{M}^{0.40}$ (with a correlation coefficient of 0.78). In comparison, Olofsson et al. (1993a) derived v_e$\propto$ $\dot{M}^{0.53}$, but their mass loss rates were calibrated using Eq. (14), which includes a v_e²-dependence. The correlation is much tighter than that obtained by Olofsson el al. (1993a), which is reassuring. Thus, the mass loss mechanism operates such that mass loss rate and expansion velocity increase together. However, the scatter appears larger than the uncertainties in the estimates, and so the mechanism also produces widely different mass loss rates for a given expansion velocity. At high mass loss rates the scatter is larger with a possible divison into objects with high mass loss rates but only moderately high velocities, and objects with moderately high mass loss rates but high velocities. We note also that for the detached shell sources the mass loss rates and expansion velocities for the two mass loss epochs follow the general trend.

Habing et al. (1994) studied the momentum transfer from the photons, via the dust, to the gas in the CSEs of AGB stars. They found that for mass loss rates close to the minimum mass loss rate for a dust-driven wind, $\sim$3 10 $^{-8}\,M_\odot$yr^-1, v_eincreases linearly with $\dot{M}$, while for mass loss rates above $\sim$10 $^{-6}\,M_\odot$yr^-1 the dependence weakens considerably. At 10 $^{-5}\,M_\odot$yr^-1 they found $v_{\mathrm e}\propto\dot{M}^{0.04}$. Habing et al. attribute the increase in v_e with mass loss rate (for low mass loss rates) to a higher efficiency in the coupling between gas and dust. This can explain the observed behaviour in Fig. 11 for mass loss rates below about 10 $^{-6}\,M_\odot$yr^-1, but for the higher mass loss rates it appears that a dependence of v_e (and $\dot{M}$) on luminosity gives the most reasonable explanation, see Sect. 6.3.

   
6.3 Dependence on stellar properties

In Fig. 12 we plot the circumstellar characteristics, $\dot{M}$ and v_e, against the stellar characteristics luminosity (L), period (P), effective temperature ( T_eff), and the photospheric C/O-ratio. The effective temperatures and C/O-ratios used in this analysis are those presented in Olofsson et al. (1993b). We have looked for dependences assuming that the ordinate scales as the abscissa to the power of $\xi$, and a normal correlation coefficient r was calculated as an estimate of the quality of the fit, see Fig. 12. The uncertainties in the estimated quantities are of the order: $\pm$50% ($\dot{M}$), $\pm$2kms^-1 ( v_e), $\pm$10% (P; but some periods may be poorly determined), a factor of 2 (L), and $\pm$200K ( T_eff).

Clearly, both the mass loss rate and the expansion velocity increase with the pulsation period of the star. There is a weaker trend with the luminosity of the star. The latter dependence is not completely independent of the former, since some of the luminosities are estimated from a P-L relation. The existence of a P-Lrelation is usually attributed to a distribution in mass (Jones et al. 1994), i.e., the higher the mass the longer the period and the higher the luminosity. When looked at in detail, at least the apparent mass loss rate dependence on period may be attributed to a change from semiregular pulsation at short periods to regular pulsations at longer periods. We may therefore infer that for the mass lass rate it is not clear whether it is the regularity of the pulsation or the luminosity that causes the increase with period. For the expansion velocity the increase with period may be a combined effect of increasing mass loss rate with period (for low mass loss rates) and an increase in luminosity for the longer periods, e.g., Habing et al. 1994 derive $v_{\rm e}\propto L^{0.35}$ in their dust-driven wind model. In addition, there is evidence of a weak trend in the sense that higher mass loss rate objects have lower effective temperatures. These correlations are all consistent with a dust-driven wind, where the pulsation may play an important role. There appears to be no correlation between the mass loss rate and the C/O-ratio, which is surprising considering that the dust-to-gas mass ratio in a C-rich CSE should be sensitively dependent on this.

The present mass loss rates for the detached shell sources are typical for their periods but somewhat low for their luminosities. However, during the formation of the dCSEs the mass loss rate must have been atypically high for the present periods and luminosities. We also find that the expansion velocities of the dCSEs lie at the very high end of expansion velocities found for other stars with the same luminosity, indicating that these stars had higher luminosities during the shell ejection.

Two other stars, SZ Car and WZ Cas, stand out in these plots. The properties of SZ Car resembles the stars with known dCSEs, but the CO line profiles give no indication of a detached shell. WZ Cas has by far the lowest estimated mass loss rate in the sample, as well as the lowest C/O-ratio, 1.01, which classifies it as an SC-star. It is also a Li-rich J-star (Abia & Isern 1997).

Figure 12: The derived mass loss rate plotted against the luminosity (L), pulsational period (P), effective temperature ( Teff), and the photospheric C/O-ratio of the star. The measured expansion velocity is plotted gainst the period and the luminosity. Sources with known detached shells are indicated with triangles (open triangles represent dCSEs, while filled ones represent aCSEs). Also shown are the power law index $\xi$, assuming the ordinate to scale with the abscissa, as well as the correlation coefficient r (the detached shell sources are not included in the fit)

6.4 Enrichment of the ISM

Carbon stars on the AGB are important in returning processed gas to the interstellar medium (ISM). The total mass loss rate of carbon stars in the Galaxy is obtained from

$$\dot{M}_{\mathrm{Gal}} = \int_0^{\infty} 2\pi R\, \mbox{$\Sigma$$<$$\dot{M}$$>$ }\, {\rm d}R$$ (15)

where $\Sigma$ is the surface density of carbon stars, <$\dot{M}$>is the mean mass loss rate of the carbon stars. According to Guglielmo et al. (1998) the infrared carbon stars have a roughly constant surface density out to the galactocentric distance of the Sun, after which it follows an exponential decline with a scale length of $\sim$2.5kpc. We approximate this by assuming a constant value, $\Sigma_0$ (the surface density in the solar neighborhood), to 10kpc and zero beyond this galactocentric distance.

Based on the complete sample, i.e., stars within 500pc from the Sun, we estimate $\Sigma_0$<$\dot{M}$> to be $\sim$1.7 10 $^{-10}\,M_{\odot}$yr^-1pc^-2, where we have included also helium. This estimate is very sensitive to the number of high mass loss rate objects found within 500pc, e.g., the high mass loss rate object CW Leo contributes almost half of our estimate of the total mass returned to the ISM. Our estimate of the rate at which matter is returned to the ISM by carbon stars is consistent with previous estimates (considering the large uncertainties), e.g., Knapp & Morris (1985) and Jura & Kleinmann (1989) derive values of $\sim$2 10 $^{-10}\,M_{\odot}$yr^-1pc^-2 and $\sim$1.5 10 $^{-10}\,M_{\odot}$yr^-1pc^-2, respectively. Using Eq. (15) the annual return of matter to the ISM in the Galaxy is estimated to be $\sim$0.05$M_{\odot}$ for the carbon stars considered here. It is quite possible that a larger mass return from carbon stars is obtained during their final evolution on the AGB, a so called superwind phase, as is indicated by the estimated values for infrared carbon stars, $\sim$0.5$M_{\odot}$yr^-1(Epchtein et al. 1990; Guglielmo et al. 1993), and PNe, $\sim$0.3$M_{\odot}$yr^-1 (Gustafsson et al. 1999). This confirms the importance of carbon stars for the cosmic gas cycle in galaxies. By comparison, high mass stars are estimated to contribute about 0.04$M_{\odot}$yr^-1(Gustafsson et al. 1999).

We estimate the contribution from carbon stars to the carbon enrichment of the ISM to be $\sim$0.5 10^-4$M_{\odot}$yr^-1during the AGB stage (if the subsequent superwind phase is included this value may increase to $\sim$5 10^-4$M_{\odot}$yr^-1). This corroborates the conclusions by Gustafsson et al. (1999) that "normal'' carbon stars are not important for the total carbon budget of the Galaxy. According to these authors the main contributors should instead be high mass stars in the Wolf-Rayet stage, annually supplying the Galaxy with $\sim$0.01$M_{\odot}$ of carbon. This is roughly what is required to produce the $\sim$10^-3 10 $^{11}\,M_{\odot}$ of carbon present in the Milky Way over a period of 10¹⁰ years.