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Astron. Astrophys. 359, 586-596 (2000)

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4. Model results

4.1. The [FORMULA]CO modelling

As explained above, the 12CO line emission is used to derive the basic parameters of the CSE. The observational constraints for each source are presented in Table 4. The main part of the observational data used here are the J[FORMULA]1[FORMULA]0 and J[FORMULA]2[FORMULA]1 spectra obtained by Olofsson et al. (1993a) using the SEST, OSO, and the IRAM 30 m telescope at Pico Veleta, Spain (some of these observations have been remade and the intensities stated in Table 4 may therefore be somewhat different from those originally given in the reference). Observations of the two lowest rotational transitions have also been performed using the NRAO 12 m telescope at Kitt Peak, USA, and of the J[FORMULA]3[FORMULA]2 line using the SEST (Schöier & Olofsson 2000). In addition, we have obtained publicly available data from the James Clerk Maxwell Telescope (JCMT) at Mauna Kea, Hawaii. The JCMT data are taken at face value. However, in the cases where there are more than one observation available, the derived line intensities are generally consistent within [FORMULA]20%. In addition, the good agreement with corresponding SEST observations lend further support for the reliability of the JCMT public data.


[TABLE]

Table 4. CO modelling results compared with observations.



[TABLE]

Table 4. (continued).
Notes:
1. This paper; 2. Schöier & Olofsson (2000); 3. Schöier & Olofsson (2000, unpublished data); 4. Olofsson et al. (1993a); 5. Olofsson et al. (1993b); 6. Olofsson et al. (1996); 7. JCMT public archive; 8. Neri et al. (1998); 9. Nyman et al. (1993); 10. Ryde et al. (1999).
a) The observed line ratios have been multiplied by 0.87 to correct for the differences in line strenghts and beam-filling factors (see text for details).


Thus, for all stars we use data from more than one 12CO transition, in some cases four, to constrain the model. The intensities and overall line shapes of the circumstellar lines produced by the radiative transfer model generally agree well with those observed. This is illustrated here for two of our sample stars CW Leo (Fig. 2) and U Hya (Fig. 3). CW Leo is a high mass loss rate Mira variable where the excitation of 12CO is dominated by collisions. U Hya, on the other hand, is a low mass loss rate object where radiation emitted by the central star plays a role in the excitation. The 12CO model results presented in this paper constitute a sub-sample of the results of an analysis of a large survey of carbon stars presented in Schöier & Olofsson (2000). The reader is referred to this paper for a detailed description of the sensitivity of the model to the various input parameters. However, we point out here that tests made by Schöier & Olofsson show that the derived mass loss rate, for the majority of objects in this study, is mostly affected by the temperature structure and not by the assumed inner radius of the shell or the luminosity of the star, i.e., collisional excitation dominates over radiative excitation. In addition, Schöier & Olofsson have tested the molecular envelope size calculations by comparing radial brightness distributions obtained from the model with those observed. It was found that the envelope sizes estimated using the results from the model by Mamon et al. (1988) are generally consistent with the observations. This is illustrated here in Fig. 2 for CW Leo where the radial brightness distribution of the 12CO(J[FORMULA]1[FORMULA]0) line emission, observed at OSO, is compared to the distribution obtained from the radiative transfer model.

[FIGURE] Fig. 2. Multi-transition 12CO and 13CO millimetre wave line emission observed towards the high mass loss rate Mira variable CW Leo. The observed spectra (histograms) have been overlayed with the model prediction (full line) using a mass loss rate of 1.5[FORMULA]10-5 [FORMULA] yr-1 and a 12CO/13CO-ratio of 50. The transition, telescope used, and the beamsize, are given for each of the observations. Also shown are the observed radial brightness distributions overlayed by the results from the model (full line). The circular beam used in the radiative transfer is indicated by the dot-dashed line. The full lines in the temperature panels give the excitation temperature of the 12CO(J[FORMULA]2[FORMULA]1) or the 13CO(J[FORMULA]2[FORMULA]1) transition. The dotted lines show the kinetic gas temperature derived from the excitation analysis. The dashed lines give the tangential optical depth, [FORMULA], of the 12CO(J[FORMULA]2[FORMULA]1) or the 13CO(J[FORMULA]2[FORMULA]1) transition

[FIGURE] Fig. 3. Multi-transition 12CO and 13CO millimetre wave line emission observed towards the optically bright carbon star U Hya. The observed spectra (histograms) have been overlayed with the model prediction (full line) using a mass loss rate of 1.0[FORMULA]10-7 [FORMULA] yr-1 and a 12CO/13CO-ratio of 25. The transition, telescope used, and the beamsize, are given for each of the observations. The full lines in the temperature panels give the excitation temperature of the 12CO(J[FORMULA]2[FORMULA]1) or the 13CO(J[FORMULA]2[FORMULA]1) transition, dash-dot lines indicate a negative excitation temperature, i.e., maser action. The dotted lines show the kinetic gas temperature derived from the excitation analysis. The dashed lines give the tangential optical depth, [FORMULA], of the 12CO(J[FORMULA]2[FORMULA]1) or the 13CO(J[FORMULA]2[FORMULA]1) transition

The derived envelope parameters, used as input for the 13CO modelling, are presented in Table 3. We believe that, within the adopted circumstellar model, the estimated mass loss rates are generally accurate to within [FORMULA]50% (neglecting errors introduced by the uncertain CO abundance and the distance estimates). In the models presented here only 12CO line cooling was included. For the J-stars, cooling from 13CO will be important (scales with the isotope ratio for these low mass loss rate objects), but it will not affect the derived mass loss rate since the heating must be increased, i.e., the kinetic temperature structure will not change significantly, in order to maintain a good fit (Schöier & Olofsson 2000). See also the discussion in Ryde et al. (1999) for the high mass loss rate object IRAS 15194-5115, where radio and far-IR 12CO and 13CO data are modelled.

When comparing our mass loss rate estimate for CW Leo to those obtained from other detailed radiative transfer models (Kastner 1992; Crosas & Menten 1997; Groenewegen et al. 1998) we find a very good agreement, within 20%, when adjustments for differences in [FORMULA] and distance have been made. Kastner (1992) also modelled the high mass loss rate object RW LMi (a.k.a. CIT 6) obtaining (when corrected for the difference in distance) a mass loss rate in excellent agreement with our estimate. Sopka et al. (1989) used a much simpler radiative transfer model to derive the mass loss rate for a number of AGB stars, of which four overlap with our survey. For RW LMi they derived a mass loss rate in excellent agreement with that obtained from our model. For LP And, V384 Per, and V Cyg, however, there are discrepancies of about a factor of two to three. It should be noted though that Sopka et al. (1989) based their mass loss rate estimates on observations of a single line, J[FORMULA]1[FORMULA]0, using a single telescope (OSO). For LP And, the source with the largest discrepancy, we observe a 12CO(J[FORMULA]1[FORMULA]0) line that is almost a factor of two stronger using the same telescope.

4.2. The [FORMULA]CO modelling

Once the general characteristics of the CSE have been determined from the 12CO analysis, the observed 13CO line emission is modelled. Again, the observed intensities, as well as the line shapes, are generally well reproduced by the model. As for the 12CO modelling, this is illustrated for two of our sample stars, CW Leo (Fig. 2) and U Hya (Fig. 3).

The abundance of 13CO is usually (the J-stars provide an exception) much smaller than that of 12CO, which leads to significantly different excitation conditions for the rarer isotopomer. For instance, the model 13CO line intensities are more sensitive to the assumed properties of the central source of emission, since radiative pumping via the first excited vibrational state becomes important. Indeed, the 13CO J[FORMULA]1[FORMULA]0 and J[FORMULA]2[FORMULA]1 transitions have inverted populations over parts of the envelope even for a high mass loss rate object as CW Leo, Fig. 2. For the high mass loss rate objects, however, the part over which there is a weak maser acting is very small and the emission emanating from this region is not detected in our observations. For thinner envelopes the part of the envelope where the lowest rotational levels are inverted is larger due to the fact that the pumping emission can penetrate further out into the wind. In the case of U Hya, changing the inner radius of the CSE or the luminosity of the star by [FORMULA]50% will change the estimated 13CO abundance by [FORMULA]20%.

In our modelling of the 13CO line emission we have assumed that the 13CO envelope size is equal to that of 12CO. This is based upon the model results by Mamon et al. (1988). In this model both the effects of photodissociation and of chemical fractionation were included. Chemical fractionation of CO occurs through the exchange reaction (Watson et al. 1976) [FORMULA]+[FORMULA]+[FORMULA]+[FORMULA], where [FORMULA][FORMULA]35 K. Below this temperature, the backward reaction is suppressed and production of 13CO from 12CO is favoured. This reaction can effectively produce 13CO in the outer, cool parts of circumstellar envelopes. Mamon et al. (1988) concluded that the difference between the 12CO and 13CO abundance distributions, when tested over a large mass loss rate interval, is always small, no more than 10 to 20%. Without the effect of chemical fractionation the 13CO envelope would be significantly smaller. This is due to the fact that CO is photodissociated in lines and thereby exhibits considerable self-shielding. Thus, the shielding is higher the higher the optical depth, and hence it is less efficient for the less abundant 13CO. Using OSO we have obtained a brightness distribution map of the 13CO(J[FORMULA]1[FORMULA]0) emission around CW Leo. It is found that the model, with an assumed 13CO envelope size equal to that of 12CO, reproduces the observed radial brightness distribution within the observational errors, Fig. 2. A 20% smaller 13CO envelope gives the same 12CO/13CO-ratio but fails to reproduce the observed radial brightness distribution within the observational errors. This shows the importance of chemical fractionation, at least in the cool outer parts of dense CSEs. In the case of the thin molecular envelope around U Hya a 20% smaller 13CO envelope size would increase the derived 13CO abundance by about 20%. This illustrates that in the case of a thin CSE the CO molecules are effectively excited to the photodissociation radius, i.e., photodissociation determines the size of the emitting region, whereas in a dense CSE excitation sets the size of the emitting region.

4.3. The circumstellar [FORMULA]CO/[FORMULA]CO-ratio

In Table 4 we list the observed integrated 12CO/13CO line intensity ratios. They are corrected for the differences in line strengths and beam-filling factors (assuming the sources to be unresolved the combined effect gives a [FORMULA] correction, i.e., the observed ratio is lowered by a factor of 0.87). Although we note that some of our CSEs are resolved, we have nevertheless treated all our sources in the same manner. Here we have chosen to integrate the emission over the entire line in order to achieve better signal-to-noise ratios. For optically thin lines this ratio should provide a first order estimate of the abundance ratio. A narrow velocity interval centered on the systemic velocity, where optical depth effects are smallest, should give somewhat higher line intensity ratios than those presented here. Nine of our sample stars have been detected in the 13CO(J[FORMULA]1[FORMULA]0) and/or the 13CO(J[FORMULA]2[FORMULA]1) line. A simple comparison of the 12CO/13CO line intensity ratios and the 12C/13C-ratios derived by Lambert et al. (1986) shows a tight correlation of the form (in the cases where more than one value is available in Table 4 we have used an average),

[EQUATION]

Thus, a straightforward use of line intensity ratios would lead to 12C/13C-ratios that, on the average, agree with those of Ohnaka & Tsuji (1996). However, any optical depth effects would lower the observed line intensity ratio.

In Table 3 we present the 12CO/13CO-ratios derived using our radiative transfer code. They span a large range, from 2.5 to 90 (steps of 5 in the 12CO/13CO-ratio was used to find the best fit 13CO model, except for the J-stars where a smaller step-size of 0.5 was used). For most of our observed stars the isotope ratio obtained from the detailed radiative transfer is higher than those estimated from simple line intensity ratios. The discrepancy is fairly small for the objects with thin CSEs, but it increases with the thickness of the CSE ([FORMULA][FORMULA]/[FORMULA]) and reaches a factor of six for the high mass loss rate objects, Fig. 4. This reflects that even for low to intermediate mass loss rate objects there are optical depth effects, and a detailed modelling is needed to derive reliable isotope ratios. In Table 3 we present, for each star, the maximum tangential optical depth in the 12CO(J[FORMULA]2[FORMULA]1) transition obtained in the modelling. The radial variation of the tangential optical depth is shown for the high mass loss rate object CW Leo in Fig. 2 and for the low mass loss rate object U Hya in Fig. 3. The optical depth in the 12CO(J[FORMULA]1[FORMULA]0) line is significantly lower.

[FIGURE] Fig. 4. The ratio between the 12CO/13CO-ratio obtained from the radiative transfer analysis and that obtained from the observed line intensity ratios (in the cases where more than one value is available in Table 4 we have used an average) as a function of the density of the shell ([FORMULA][FORMULA]/[FORMULA]). Sources with detached CSEs are marked as open circles.

The estimated 13CO abundance depends on the adopted 12CO abundance (assumed to be equal for all stars). We find that to a first approximation it scales with [FORMULA], since a lower (higher) [FORMULA] leads to a higher (lower) mass loss rate to fit the 12CO line intensities and hence a lower (higher) 13CO abundance to fit the 13CO line intensities. This means that to a first approximation the estimated 12CO/13CO-ratios are only very weakly dependant on the adopted 12CO abundance. We have varied some of the other parameters in the model (see above), and conclude that in doing so the derived 12CO/13CO-ratio changes by about 20%. With an additional uncertainty of [FORMULA]15% in the relative calibration of the 12CO and 13CO data (the relative calibration is usually better than the absolute calibration for any given telescope), we estimate that the 12CO/13CO-ratio is uncertain by about [FORMULA]30%. In the cases were the 12CO/13CO-ratio estimate relies on observations of just one line the uncertainty may be as high as [FORMULA]50% depending on the quality of the data.

We compare first our derived 12CO/13CO-ratio for CW Leo with those found by others. Crosas & Menten (1997) derived, using a radiative transfer model similar to ours, an isotope ratio of 50, i.e., the same as we do. Greaves & Holland (1997) found a lower ratio of 32, and also a much lower ratio (24) for LP And, using a simple radiative transfer model. Kahane et al. (1992) found the 12C/13C-ratio to be 44 based on observations of optically thin lines. Kahane et al. (1992) also determined the 12C/13C-ratio for RW LMi obtaining a value of 31, in good agreement with our 12CO/13CO estimate for this object. Sopka et al. (1989) estimated 12CO/13CO-ratios for four of our stars that are all lower than our derived values. This is most probably an effect of the difference in the treatment of the radiative transfer. We also note that Dufour et al. (2000, in prep.), derive somewhat lower 12CO/13CO-ratios for the three J-stars.

For our sample stars we compare the circumstellar 12CO/13CO-ratios obtained from the modelling with the photospheric 12C/13C-ratios estimated by Lambert et al. (1986) in Fig 5. A good correlation of the form

[EQUATION]

is obtained. Lambert et al. (1986) give an uncertainty in their estimated 12C/13C-ratios of about [FORMULA]40%. Included in our sample are stars that have had a drastic change in their mass loss rate. For R Scl and S Sct, stars with known detached CSEs (probably) produced during a period of intense mass loss, we have performed the analysis using the envelope parameters determined by Olofsson et al. (1996). These results are included in Fig 5. We note here that our results suggest that the 12C/13C-ratios in the detached shells (with ages of about 103 and 104 yr for R Scl and S Sct, respectively) are the same as the present ones in the photospheres.

[FIGURE] Fig. 5. Comparison between the estimated circumstellar 12CO/13CO-ratio obtained using our detailed radiative transfer code, and the photospheric 12C/13C-ratio estimated by Lambert et al. (1986). An open triangle indicate a lower limit and the solid line represents a 1:1 correlation. Note that at 12C/13C[FORMULA]3 there are results for three stars.

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Online publication: July 7, 2000
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