6 Conclusions

We have shown that the careful analysis of multi-line single dish observations with a relatively large beam can provide a set of information comparable to single-line interferometric observations. From a careful excitation analysis using a self-consistent radiative transfer computation it is possible to deduce some sub-resolution information. We can infer clump sizes, masses and densities at scales below a tenth of the beam size. However, interferometric observations are necessary to determine the exact core geometry including the location and number of clumps within a dense core.

The spherically symmetric radiative transfer code used here is able to take into account radial gradients in all quantities and internal clumpiness of the cloud. It enables a reliable deduction of the physical parameters from line profiles observed in sources with a size close to the spatial resolution limit. The method allows to analyse similar observations of objects like star-forming cores in distant galaxies unresolvable by all today's means. For a better resolution of the internal temperature structure the approach should be combined with sophisticated models on the energy balance including the continuum radiative transfer in the future. Although the simple escape probability analysis gives a reasonable estimate for the column density, it fails regarding the density and temperature structure.

The line analysis shows two essential points:
i) The main constraints on the structural quantities which can be deduced from the observations are set by the tracer. The range of densities and temperatures that one can determine from the radiative transfer calculations is restricted by the transitions observed. In case of the CS 2-1, 5-4, and 7-6 lines, the covered densities range from about $2\times10^5$ to 10⁷ cm^-3. With additional information from the CS 10-9 the upper limit can be extended by another factor 5. The information from the rarer C³⁴S isotope cannot extend the density interval but reduces the error bars and provides better estimates for the clumpiness of the medium. The high resolution observations discussed in Sects. 3.2 and 5.4 show that different tracers provide access to different types of information whereas the parameters from our CS observations agree well with the CS results there.
ii) Temperature and clumpiness are related quantities. When turbulent clumping in the cloud is neglected, the temperature determination will necessarily fail. On the other hand does accurate information on the clumpy structure of a massive core help to constrain the temperature structure. Additional observations in higher transitions or complementary estimates of the clumpiness will help to reduce the uncertainty of the temperatures. Thus spatial resolution is still essential. For nearby clouds we get a good agreement with results from other high-resolution observations, but for distant massive cores the temperature structure is still an open question.

All massive cores that we have analysed are characterised by turbulent clumpiness with typical clump sizes of 0.01-0.02 pc. The clouds are approximately virialised and show density gradients around -1.6 but with a scatter between -1.1 and -2.2. Large parts of the cores follow a constant temperature, but we must admit a considerable uncertainty in the most inner and outer parts. The correlation between the cloud temperature and the turbulent line width indicates that related processes should be responsible for heating and turbulent driving.

Future observations of dense cores should focus on different tracers to gain access to additional information which cannot be deduced from a single tracer such as CS. As a drawback, the full uncertainty of today's chemical models will enter and partially limit the interpretation of the observations.

Acknowledgements

We thank J. Howe for providing us with the CS 2-1 observational data. We are grateful to the anonymous referee for many detailed comments helping to improve the paper considerably. This project was supported by the Deutsche Forschungsgemeinschaft through the grant SFB 301C. The KOSMA 3m radio telescope at Gornergrat-Süd Observatory is operated by the University of Cologne and supported by the Land Nordrhein-Westfalen. The receiver development was partly funded by the Bundesminister für Forschung and Technologie. The Observatory is administered by the Internationale Stiftung Hochalpine Forschungsstationen Jungfraujoch und Gornergrat, Bern. The research has made use of NASA's Astrophysics Data System Abstract Service.