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Astron. Astrophys. 353, 583-597 (2000)

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4. The shell emission

In this section we present the observational results for the shell emission. The analysis follows to a large extent that used in Olofsson et al. (1996, 1998). We also estimate the mass and the physical properties of the shell gas, as well as discuss the density distribution within the shell.

4.1. Comparison with single-dish fluxes

We will here estimate whether some of the total CO emission is missing in the interferometer data [see e.g. Adler et al. (1992) for a discussion of this]. We have computed the CO fluxes, integrated over the brightness distributions in four velocity ranges [the full velocity range (-40[FORMULA]/-14.5[FORMULA]), a velocity range centred on the systemic velocity (-29.5/-25.5), and a blue- (-40.5/-37.5) and a redshifted (-17.5/-14.5) velocity interval], using our interferometer data and the single-dish data obtained by Olofsson et al. (1996) with the IRAM 30 m telescope. The results are summarized in Table 1. For the J=[FORMULA] line the total interferometer flux amounts to [FORMULA]85% of the total single dish flux, i.e., within the calibration uncertainties it is possible that the fluxes agree, and hence very little extended emission is missed by the interferometer. However, there is a clear trend over the line. At the line centre the interferometer flux has its maximum relative to the single-dish flux ([FORMULA]90%). This points towards an emission that is not more widely distributed in the radial direction than what is seen in the interferometer maps close to the systemic velocity, i.e., all the CO emission comes from a narrow shell. If we assume that no flux is missed by the interferometer at the line centre (i.e., the interferometer fluxes are scaled by a factor 1/0.9 to correct for calibration differences between the interferometer and the single-dish data), we conclude that [FORMULA]40% ([FORMULA]25%) of the extreme blue- (red-) shifted emission, where the emitting regions are two orders of magnitude larger than the FWHM of the synthesized beam, may be resolved out by the interferometer. The shortest projected baseline corresponds to an angular resolution of [FORMULA]30" and [FORMULA]15" in the J=[FORMULA] and J=[FORMULA] lines, respectively. In the J=[FORMULA] line this is comparable to the sizes of the emitting regions at the extreme velocities, and so some of the missing flux may come from extended emission. It is possible that part of the effect at the extreme velocities is also due to the difficulties in CLEANing the dirty images in these velocity ranges where the emission is very extended. In conclusion, these results are consistent with the CO emission coming mainly from a geometrically thin shell consisting of a clumped medium.


Table 1. Comparison between shell fluxes [integrated over the shell brightness distributions in four velocity ranges: the full velocity range (-40[FORMULA]/-14.5[FORMULA]), a range centred on the systemic velocity (-29.5/-25.5), and a blue- (-40.5/-37.5) and a redshifted (-17.5/-14.5) velocity interval] obtained from single-dish and interferometry data

The J=[FORMULA] interferometer map does not cover the entire shell (in some velocity ranges it is missed altogether), and we can only make a crude comparison for the emission around the systemic velocity. Only about 45% of the single-dish flux is present in the interferometer data. This can be partly attributed to the limited coverage of the shell. About 20% of the emitting area is not covered, see Fig. 2, and hence in a large fraction of the shell at least 65% of the J=[FORMULA] single-dish flux is present in the interferometer data. Considering the results for the J=[FORMULA] data and the resolution obtained in the J=[FORMULA] line for the shortest baseline, we would expect only little flux ([FORMULA]10%) to be missed at the systemic velocity in the J=[FORMULA] data. Therefore, there is possibly a discrepancy in the absolute calibration of the interferometer and single-dish data.

4.2. The global shell geometry and structure

The impression obtained from Fig. 1 is that the emission arises in a very thin shell with a high degree of circular symmetry. This is emphasized in Fig. 4 where a circular ring, with a Gaussian intensity distribution in the radial direction, has been fitted to the J=[FORMULA] data in the -27.5[FORMULA]2[FORMULA] interval. The radius of the ring is 34[FORMULA]7 and its full width at half maximum is 3[FORMULA]2 (when deconvolved with a 2" beam the shell width is 2[FORMULA]5). However, the centre of the ring is offset 1[FORMULA]8 [PA[FORMULA]-16o; in the following the position angle (PA) is counted from the north to the east] with respect to the position of the star (here given by the peak position of the central emission). We will analyze this offset further in Sect. 4.4.

[FIGURE] Fig. 4. A fit of a circular, Gaussian ring to the CO(J=[FORMULA]) data in the -27.5[FORMULA]2[FORMULA] interval. The centre of the ring is marked with a +, and the position of the star (here given by the peak position of the central emission) with an *. The contours lie at 30, 60, and 90% of the maximum intensity

We have also fitted circular, Gaussian rings to all the J=[FORMULA] brightness maps shown in Fig. 1, in order to study the 3D-geometry of the shell. There are only small deviations from circular symmetry in all velocity intervals. The results are shown in Fig. 5, where the ring radius is plotted versus the line-of-sight velocity. The best fit of the relation [FORMULA] is also shown. The shell is expected to obey this relation if it is spherically symmetric and expands with the same velocity in all directions, in which case [FORMULA] is the systemic velocity and [FORMULA] is the gas expansion velocity. An excellent fit is obtained for [FORMULA]=34[FORMULA]78[FORMULA]0[FORMULA]03, [FORMULA]=-27.33[FORMULA]0.01[FORMULA], and [FORMULA]=12.58[FORMULA]0.01[FORMULA] (the errors only indicate the goodness of the fit to the data points). This strongly suggests that the shell has a remarkable overall spherical symmetry. The linear average shell radius and width are [FORMULA]2.7[FORMULA]1017 cm and [FORMULA]1.9[FORMULA]1016 cm, respectively. The dynamical time scales, [FORMULA]=r/[FORMULA], for the shell radius and width are [FORMULA]6800 yr and [FORMULA]500 yr, respectively. We will adopt -27.3[FORMULA] and 12.6[FORMULA] as the systemic and expansion velocities of the shell, respectively (thus, the maps in Figs 2 and 4 are all centred within 0.2[FORMULA] of the estimated systemic velocity). The systemic velocity estimated from the central emission agrees exactly with the result for the shell.

[FIGURE] Fig. 5. The estimated shell sizes as a function of line-of-sight velocities for the CO(J=[FORMULA]) data, and the best fit of the relation [FORMULA] to these data (see text for details). The 1[FORMULA] errors of the size estimates lie in the range 0[FORMULA]01 (at the systemic velocity) to 0[FORMULA]08 (at the extreme velocities)

For a thin, approximately spherical shell that expands with a constant velocity a relatively accurate reconstruction of the 3D-geometry may be obtained. We have used the CO(J=[FORMULA]) data to produce Mollweide projections of the shell radius (here defined as the distance of the shell to its centre), the shell width, and the observed brightness, Fig. 6. The shell centre is obtained from the fit shown in Fig. 4. In the final stage the results are smoothed to a resolution of [FORMULA]10o on the sphere. This is done to minimize effects due to the limited angular and spectral resolutions of the observations. The effect of the interferometer missing flux from extended emission is most severe in the regions where the gas moves directly towards (the extreme blue-shifted emission) and away (the extreme red-shifted emission) from us (see Sect. 4.1), and we have therefore blanked the results in two cones, with opening angles of 60o, directed towards and away from us. In principle, parts of the structure in Fig. 6 could be due to an expansion velocity that varies with direction, and hence causing an error in the deprojection. It is difficult to estimate whether or not this is the case. However, we find that e.g. the variations in the estimated shell radius over the sphere are very similar to those estimated from the data close to the systemic velocity (i.e., where projection effects play a very limited rôle, see Fig. 10). Thus, we believe that expansion velocity variations have a negligable effect on the results in Fig. 6.

[FIGURE] Fig. 6. Top: The distance of the shell to its centre, in units of the average distance, as estimated from the CO(J=[FORMULA]) data (upper). Middle: The width of the shell, in arc seconds, as estimated from the CO(J=[FORMULA]) data (middle). No correction for the beam smearing has been done. Bottom: The observed CO(J=[FORMULA]) brightness, in Jy beam-1, of the shell (lower). The results of the deprojection are smoothed to a resolution of 10o on the sphere (see text for more details). The results in cones of opening angle 60o directed towards and away from us have been blanked

The results presented in Fig. 6 emphasize the overall spherical symmetry. The shell radius is calculated as the intensity-weighted distance. All distances lie within [FORMULA]3%, i.e., within [FORMULA]1", of the average distance 34[FORMULA]8. There are no apparent trends, but there are regions where the shell is systematically curved inwards or outwards.

The width is obtained from the standard deviation of the distance estimate, but given in terms of a full width at half maximum of a Gaussian distribution. This should not be taken as the true shell width that would be obtained from a Gaussian fit in the radial direction, but the result reflects any trends there are. The broadening due to the [FORMULA]2" beam has not been removed. There is a clear large-scale trend in the sense that the shell is markedly thicker towards its northern pole.

The observed CO(J=[FORMULA]) brightness of the shell (note that this is not the surface brightness that would be seen from directions perpendicular to the shell surface) is patchy with a variation on a large angular scale by at least a factor of three between different regions. In particular, the lower hemisphere is markedly brighter than the upper one. Thus, the brightness decreases in this region where the shell broadens. It therefore seems that the variation in the total emission is more limited. We have used the Gaussian fits to the radial J=[FORMULA] intensity distributions (in the -27.5[FORMULA]2[FORMULA] interval) presented below in Fig. 9, to show that the product of the intensity and width varies considerably less with position angle than the deconvolved width and somewhat less than the intensity, Fig. 7. Even though the width of the shell varies by about a factor of three and the brightness by about a factor of two, the variation in the total intensity is rather moderate (the standard deviation is 0.15 for the normalized product). This may suggest that the amount of material in the shell is rather evenly distributed on the larger scale (i.e., at least when averaged over solid angle intervals of about 0.2 steradians). Of course, one should caution here that there is not necessarily a one-to-one relationship between brightness and density distribution, e.g., due to saturation. On the other hand, there are several processes that tend to make the brightness distribution have more contrast than the density distribution, e.g., photodissociation and excitation.

[FIGURE] Fig. 7. The normalized (with the respective average values) intensities, deconvolved widths, and products of intensity and width obtained from the Gaussian fits to the CO(J=[FORMULA]) data presented in Fig. 9. Note that the data in the 0-90o interval are repeated at the end

We will finally analyze in yet another way the global distribution of the CO line-emitting material in the shell. Each map in Fig. 1, which covers emission from a 1[FORMULA] interval, contains emission from 1/2[FORMULA][FORMULA]4% of the volume of the shell (for a spherical shell of radius R and width [FORMULA] that expands with the velocity [FORMULA] the volume emitting radiation in the narrow velocity range [FORMULA] around the line-of-sight velocity [FORMULA] is given by 2[FORMULA][FORMULA][FORMULA][FORMULA]/[FORMULA], i.e., independent of [FORMULA], and the full velocity width of the emission is 2[FORMULA]). Thus, we expect each map to contain the same amount of emission if the emitting material is uniformly distributed on the large scale and the emission is optically thin. Similarly, if the emission is optically thick but the medium consists of a large number of small clumps that are uniformly distributed (with only limited amounts of shadowing), each map will contain the same amount of emission. If the emission is optically thick and the medium is smoothly distributed we expect the emission to decrease towards the systemic velocity. In Fig. 8 we show the product of the shell brightness, the shell width, and the shell size for each map (obtained from the fits of Gaussian circles), i.e., a measure of the total emission from each map. Except for the two extreme velocity maps (where the effects of resolution on the interferometer fluxes are expected to be present) the product is essentially constant. This excludes a smooth, optically thick medium, and points to optically thin emission or, more likely considering the discussion in Sect. 4.1, a highly clumped medium that may be optically thick. Furthermore, averaged over the scales of the maps in Fig. 1 the CO gas appears to be evenly distributed over the shell. The constant product, which is lower by about 50% only at the two extreme velocities, also strengthens the above conclusion that resolution effects in the interferometer data are quite limited.

[FIGURE] Fig. 8. The products of the intensities, widths, and sizes, obtained from the fits to the CO(J=[FORMULA]) shell emission, plotted against the line-of-sight velocity (the results are normalized to the value for the -27[FORMULA] map)

We will end this section by examining the CO brightness distribution across the shell, i.e., the radial CO brightness distribution. Once again, we use the J=[FORMULA] data in the -27.5[FORMULA]2[FORMULA] interval, for which we have estimated that very little, if any, of the flux is missed by the interferometer. The data are averaged over position angle intervals of 30o and then presented in 0[FORMULA]5 bins in Fig. 9. Obviously, there is also an averaging along the line-of-sight. These radial brightness distributions are essentially perfectly fitted by Gaussians in all PA-intervals. That is, although the shell is resolved (marginally in some PA-intervals), we find no clear evidence for any structure in the radial brightness distribution, e.g., an edge brightening. The radial brightness distributions also show low-intensity large-scale undulations that must be attributed to the problem of CLEANing such a complicated image. However, this also puts limits on any low-intensity structure that can be identified. Thus, we cannot confirm the indications of weak J=[FORMULA] emission on the inside of the shell that was reported by Olofsson et al. (1998) for a limited PA-interval. Here we conclude that any emission immediately inside or outside the shell is weaker than [FORMULA]5% of the shell peak brightness when azimuthally averaged over the 0o-360o interval.

[FIGURE] Fig. 9. The radial CO(J=[FORMULA]) brightness distributions, in 0[FORMULA]5 bins, averaged over position angle intervals of 30o and integrated over the velocity interval -27.5[FORMULA]2[FORMULA], starting with the 0o-30o interval at the top to the 330o-360o interval (alternating filled and open circles, with the curves offset by 0.2 Jy beam-1 km s-1 with respect to each other). The lowest curve shows the result for the entire 0o-360o interval (filled squares). The radial distance is measured from the estimated shell centre

4.3. Shell radius, width and CO brightness variations

It has been shown in the preceding section that the shell has an almost perfect overall spherical symmetry, and that the CO(J=[FORMULA]) emission is relatively uniformly distributed over the shell when averaged over sufficiently large solid angles (about 0.2 steradians). We shall in this section analyze the radius, the width, and the CO brightness of the shell in more detail, although using data from only parts of the shell. The best estimates of the radius and width are obtained from the maps centred around the systemic velocity. It appears that very little of the total flux is missed by the interferometer at these velocities, see Sect. 4.1. We will use the results from the Gaussian fits to the radial J=[FORMULA] brightness distributions presented in Fig. 9. We have done the same thing for the J=[FORMULA] data in the -27.5[FORMULA]2[FORMULA] interval and in the PA-intervals where reliable estimates can be obtained. We caution once again that the J=[FORMULA] data may miss significant amounts of flux, something which may affect, in particular, the width estimates. The results are summarized in Fig. 10.

[FIGURE] Fig. 10. The shell radius and width (deconvolved full width at half maximum), estimated from the CO(J=[FORMULA] and J=[FORMULA]) data in the -27.5[FORMULA]2[FORMULA] interval, are shown as a function of the position angle. Note that the J=[FORMULA] data in the 0-90o interval are repeated at the end

The radii estimated from the J=[FORMULA] and J=[FORMULA] data agree almost perfectly. There are clear smooth variations in the shell radius, but the difference between the maximum and minimum radius is only 1[FORMULA]2. Likewise, the widths (deconvolved full width at half maximum) estimated from the J=[FORMULA] and J=[FORMULA] data agree reasonably well. There is a considerable variation in the shell width (in the J=[FORMULA] data that cover the full PA-range), with a minimum of [FORMULA]1[FORMULA]3 at [FORMULA]PA[FORMULA]=75o and a maximum of [FORMULA]3[FORMULA]7 at [FORMULA]PA[FORMULA]=0o, i.e., the width varies between 1 and 3[FORMULA]1016 cm, and the corresponding time scale range is 250 to 750 yr provided that the gas expansion velocity can be used in estimating this. The width/radius ratio ranges from [FORMULA]0.04 to 0.1. There is a trend that the shell distance to the centre is larger in those areas where the shell is geometrically thicker.

The detailed brightness distribution is most clearly seen in the maps obtained close to the extreme velocities, where the line-of-sight is almost in the radial direction through the shell. In order to avoid as much as possible resolution effects we have chosen to present the J=[FORMULA] map centred at -38[FORMULA], Fig. 2. This map contains emission from [FORMULA]4% of the shell volume (see above). Using the same line of arguments as that in Sect. 4.1 we estimate that it contains [FORMULA]60% of the single-dish flux from this region. Thus, we can expect that there is only a limited enhancement of the small-scale structure in the interferometer data. The brightness distribution is very patchy, and there are regions with only very weak emission. The size scale appears to be set largely by the resolution, suggesting that there is unresolved emission. We will discuss this suggested clumpy structure further in Sect. 4.5.

4.4. The shell centre

The centre of the Gaussian ring fitted to the J=[FORMULA] data in the -27.5[FORMULA]2[FORMULA] interval (Fig. 3) is offset 1[FORMULA]8 (PA[FORMULA]-16o) with respect to the peak position of the central J=[FORMULA] emission. The latter agrees well with the Hipparcos position of the star. In Fig. 11 we show the results of fits of rings to all the J=[FORMULA] brightness maps of Fig. 1. The result is a roughly constant offset by [FORMULA]1[FORMULA]7 at a PA[FORMULA]-20o in the velocity range -27.5[FORMULA]9.5[FORMULA]. Outside this range the offsets and the position angles start to vary, but this is also the regions where the emission becomes very extended and the uncertainties in the fits increase significantly.

[FIGURE] Fig. 11. The offsets to the position of the central emission of the centre of the rings, fitted to the CO(J=[FORMULA]) data shown in Fig. 1. The sizes of the offsets are denoted by filled circles, and the position angles by crosses

The essentially circular form of the brightness distributions, and the roughly constant offset as a function of the line-of-sight velocity (and hence the distance along the line-of-sight), suggest to us that the offset between the star and the shell centre is an effect of a relative motion between the two, rather than due to asymmetric expansion of the shell. Thus, the reason for the offset is probably not a consequence of the ejection nor of an interaction with an asymmetric or moving surrounding medium. We note that the average line-of-sight velocity of the interstellar molecular medium is [FORMULA]10[FORMULA] in the direction of TT Cyg, i.e., very different from the velocity of TT Cyg). Thus, effects of interaction with the ISM cannot be entirely excluded, in particular since the shift is towards the north where the shell seems to be breaking up. On the other hand, one would have expected detectable departures from circular symmetry, since in this case the shell must expand [FORMULA], or by 10% of the full expansion velocity, faster towards the north. Also, a deformation of the shell should cause a trend in the offsets along the line-of-sight. Finally, TT Cyg has a proper motion of 6.3[FORMULA]1.0 mas yr-1 (PA[FORMULA]-120o) as measured by Hipparcos. This means that the star and the shell have moved [FORMULA]40" in the tangential direction since the ejection of the shell (assuming constant expansion velocity), and hence that any possible interaction between the shell and the surrounding medium is likely to be very weak. We have not detected any interstellar CO emission in the direction of TT Cyg. An upper limit on the density of the interstellar medium of [FORMULA]0.03 cm-3 is required to have a mean free path longer than 3[FORMULA]1017 cm (corresponds to [FORMULA]40" at the distance of TT Cyg). Nevertheless, a reasonable explanation to the large scale variation of the shell width may be some kind of interaction with surrounding gas, albeit a fairly weak one.

If the offset is interpreted as due to relative motion between the shell and the star the tangential velocity difference between the star and the shell can be estimated to be [FORMULA]. This result is independent of the distance, but assumes that the shell expansion velocity can be used to estimate the time scale (we find no difference in the systemic line-of-sight velocities of the centre and the shell emission, but the uncertainties of these estimates, in particular that of the centre emission, is estimated to be at least 0.5[FORMULA]). Such a velocity difference could be due to the star being a member of a binary system. In all likelihood the mass of the companion is lower than that of TT Cyg (a less massive companion would be on the main sequence, while a more massive companion should have reached the white dwarf stage). Very crudely, assuming that TT Cyg and its companion both have a mass of [FORMULA]1[FORMULA], the observed velocity of TT Cyg suggests a wide binary with a semi-major axis of [FORMULA]1300 AU, or [FORMULA]3" (the distance between the two stars decreases if the companion is less massive than TT Cyg). If the companion is still on the main sequence its luminosity would be comparable to, or less than, that of the Sun, and a star with a solar luminosity would have a visual magnitude of [FORMULA]13, which would be difficult to detect in the vicinity of the very bright carbon star ([FORMULA]). However, towards the blue the brightness contrast improves in favour of the warmer star. The companion should be found along a line with a PA[FORMULA]70o. The estimated orbital velocity is so low that any motion of TT Cyg during the shell ejection will have negligable effects on the shell structure. Likewise, the binary system is so wide that it should have no effect on the geometry of the mass loss.

4.5. The properties of the molecular medium

We will make a crude estimate of the molecular mass of the shell and its physical state, assuming uniform density and temperature in the shell. In this case the derived results depend to a large extent on the assumed CO abundance. The CO column density is partly constrained by the observed intensity, and hence gives the H2 density, [FORMULA] (for a given shell width). Likewise, [FORMULA] and the gas kinetic temperature, [FORMULA], are jointly constrained by the line intensity ratios, and hence this determines [FORMULA]. We have used the same radiative transfer model as referred to in Sect. 3, and adopt [FORMULA]=10-3 and a shell width of 2[FORMULA]1016 cm. The density and temperature are not well constrained by the observational data. We use our J=[FORMULA] and J=[FORMULA] spectra obtained towards the shell centre with the IRAM 30 m telescope (Olofsson et al. 1996), as well as a J=[FORMULA] spectrum, obtained by us at the Caltech Submillimeter Observatory, with [FORMULA][FORMULA]0.25 K at the extreme velocity peaks. Acceptable fits to the observational data can be obtained for a range of values where the product [FORMULA][FORMULA] is roughly constant. Low and high temperature solutions are excluded, but good fits are obtained in a region centered on [FORMULA][FORMULA]250 cm-3 and [FORMULA][FORMULA]100 K. This results in a shell mass of [FORMULA]0.007[FORMULA]. The density and the temperature estimates scale roughly as [FORMULA] and [FORMULA], respectively, and the shell mass roughly as [FORMULA] (D is the distance). The emission is optically thick; the tangential optical depths are about 1.5 and 2.5 in the J=[FORMULA] and J=[FORMULA] lines, respectively, a result inconsistent with the fact that the emission in each 1[FORMULA] map is essentially the same, see Sect. 4.2. This suggests a clumpy medium. The kinetic temperature is very high at this distance from the star suggesting either a lower [FORMULA] or an efficient heating mechanism. However, if the medium is clumped the data can be fit with considerably higher densities and consequently lower temperatures. In the homogeneous model the estimated CO and H2 column densities are not enough to protect the CO molecules from photodissociation; the photodissociation time scale is [FORMULA]103 yr using the results of van Dishoeck & Black (1988) for an average interstellar UV field [to reach a photodissociation time scale of [FORMULA]104 yr requires a CO (H2) column density of [FORMULA]6[FORMULA]1016 cm-2 ([FORMULA]6[FORMULA]1019 cm-2), and consequently a molecular shell mass of [FORMULA]0.1[FORMULA]]. This again suggests a clumped medium. A mass estimate for a clumped medium will depend on the clump sizes and their temperatures (Olofsson et al. 1996). At present there exists no acceptable model from which these quantities may be derived. Our estimate of 0.007[FORMULA] for the shell mass may be a severe underestimate due to the neglect of the clumpiness in the model.

However, we can use the data to make a schematic estimate of the clumpiness of the shell molecular gas. The essentially constant total emission from the 1[FORMULA] maps and the morphology of the J=[FORMULA] brightness distributions at the extreme velocities suggest that the gas density distribution must, to a large extent, be clumpy on a size-scale less than [FORMULA]2", see Sects 4.2, and 4.3. To this we can add the above results from the smooth shell model. The single-dish data further suggest that the emission is optically thick, since in the central velocity range the J=[FORMULA] flux is about four times higher than the J=[FORMULA] flux (see Table 1), i.e., what is expected from blackbody radiation in the Rayleigh-Jeans limit. It is difficult to estimate the individual clump brightnesses, but the data presented in Fig. 2 indicate an average peak clump brightness of [FORMULA]0.05 Jy beam-1. Assuming the clumps to have Gaussian brightness distributions with FWHM:a of 0[FORMULA]5 and 2[FORMULA]0, we estimate that each clump should produce a J=[FORMULA] main beam brightness temperature of [FORMULA]0.01 and [FORMULA]0.02 K, respectively, in the IRAM 30 m telescope (the brightness temperatures are [FORMULA]20 and [FORMULA]2 K, respectively). We also estimate that each clump contributes this intensity over a velocity range less than 1[FORMULA], based on the facts that the full widths at half maximum of the extreme spectral features in the single-dish spectra are [FORMULA]1[FORMULA] (Olofsson et al. 1996), and that there is very little spatial overlap between the 1[FORMULA] maps outside the systemic velocity. The total integrated J=[FORMULA] intensity of the shell emission is [FORMULA]80 K[FORMULA] in the IRAM 30 m telescope (in the main beam brightness scale), and therefore [FORMULA]7300 and [FORMULA]3800 clumps are required to produce this emission if the clump sizes are 0[FORMULA]5 and 2[FORMULA]0, respectively. In terms of number of clumps per 1[FORMULA] interval this corresponds to [FORMULA]290, and [FORMULA]150 clumps. These estimates are lower limits since we assume no overlap in the calculation of the total emission. The volume filling factors are [FORMULA]0.01 and [FORMULA]0.4 for the 0[FORMULA]5 and 2[FORMULA]0 clumps, respectively. Thus, the appearance of the brightness maps should be very different in the two cases, and this can be used to put some further constraints on the clump properties.

We have calculated the observed brightness distributions from a number of randomly distributed "Gaussian" clumps (with an optical depth of one through the centre in order to allow for some shadowing) within a shell such that Gaussian rings fitted to these artificial brightness distributions give the same results as the fits to the observed data. We have done this comparison for the -27 and -38[FORMULA] maps, and the results for clump sizes of 0[FORMULA]5 and 2[FORMULA]0 are shown in Fig. 12. The peak brightness temperatures (in the J=[FORMULA] line) of the 0[FORMULA]5 and 2[FORMULA]0 clumps are [FORMULA]12 and [FORMULA]3 K, respectively. It is clear that the brightness distributions produced by a larger number of small clumps have a morphology that resembles that of the observed data best.

[FIGURE] Fig. 12. Artificial brightness distributions compared with the observed -27 and -38[FORMULA] CO(J=[FORMULA]) maps (middle panels). The artificial -27[FORMULA] (-38[FORMULA]) maps are produced by 180 clumps of size 2[FORMULA]0 distributed in a 0[FORMULA]8 (7[FORMULA]3) broad ring (left panels), and 500 clumps of size 0[FORMULA]5 distributed in a 3[FORMULA]1 (7[FORMULA]5) broad ring (right panels)

Using the 0[FORMULA]5 clump model and the derived shell mass we can make some crude estimates. The clump mass is of the order 10[FORMULA], i.e., [FORMULA]1 [FORMULA]. The average H2 density in a clump is [FORMULA]104 cm-3 (this also results in a considerably lower kinetic temperature, [FORMULA]20 K, required to fit the observational data). The H2 and CO column densities through a clump (diameter 8[FORMULA]1015 cm) are [FORMULA]6[FORMULA]1019 and [FORMULA]6[FORMULA]1016 cm-2, respectively. Using these values and the results of the photodissociation model of van Dishoeck & Black (1988), we estimate a photodissociation time scale for CO of [FORMULA]104 yr, i.e., as opposed to the case for the homogeneous shell this is enough to protect the molecules for a time scale comparable to the shell age, a conclusion reached also in Bergman et al. (1993) where the photodissociation was treated in some detail. This provides another strong argument for a highly clumped medium.

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Online publication: December 17, 1999