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Astron. Astrophys. 358, 682-688 (2000)

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1. Introduction

Observations of low-J 12CO and 13CO [FORMULA], [FORMULA] and [FORMULA] emission lines in several star-forming molecular cloud complexes (Castets et al. 1990; Sakamoto et al. 1994; Röhrig et al. 1995; Kramer et al. 1996; Schneider et al. 1998; Wilson et al. 1999) show that the measured line intensities and ratios cannot be produced in clouds of uniform gas temperature and density.

Typically the [FORMULA]/[FORMULA] intensity ratios for both 12CO and 13CO are close to unity. This appears to indicate optically thick thermalized emission in a homogeneous cloud for two reasons. First, if the emission were optically thin then the line ratios would be significantly greater than [FORMULA] because of the smaller optical depth of the [FORMULA] transitions. Second, if the total hydrogen density in the emitting regions were smaller than the critical density required to thermalize the populations in the [FORMULA] levels then the line ratios would become smaller than 1 due to the smaller critical densities required for thermalization of the [FORMULA] levels. However, the isotopic 12CO/13CO ratio of [FORMULA] and [FORMULA] ranges from 2 to 8. For homogeneous media this indicates optically thin rather than optically thick 13CO emission. Thus, the various CO emission lines cannot be produced in homogeneous media of uniform temperature and density. Similar results have been found in several starburst galaxies (Aalto et al. 1995; Papadopoulos & Seaquist 1998) where the isotopic 12CO/13CO ratios are even larger ([FORMULA]). Castets et al. (1990) noted that external heating with the corresponding inside-out temperature gradient can explain the observed line ratios. However, as we discuss below, the relatively narrow range of observed line ratios, as discussed below, however, requires a relatively universal astrophysical scenario for this external heating.

Observations of C+ and C18O in these star-forming regions have shown (cf. Stutzki et al. 1988; van der Werf et al. 1995) that the clouds are not homogeneous, but are in fact clumpy and consist of at least two components with high density contrast. In a clumpy environment the higher density condensations, or clumps, are illuminated by stellar FUV radiation which penetrates and is scattered through the lower-density interclump medium. Dense photon dominated regions (PDRs) (cf. Sternberg & Dalgarno 1995) form on the surfaces of the clumps. In such clumps the gas temperature decreases from the FUV heated surface layers to the colder shielded cores. As discussed in detail by Störzer et al. (1996), because of the temperature gradients, as well as geometrical effects (e.g. clumps will appear smaller in optically thinner 13CO lines compared to optically thicker 12CO lines) both the absolute and relative intensities of 12CO, 13CO and C18O emission lines will differ considerably from those produced in homogeneous clouds and plane parallel geometries.

Recent studies have shown that the observed complex structure of the molecular cloud emission can be well fitted by a decomposition of the cloud into many clumps. Typically, the clumps identified follow a power law mass distribution and mass-size relation (Kramer et al. 1998; Heithausen et al. 1998). Alternatively, the structure can be characterized as fractal. Stutzki et al. (1998) recently demonstrated that these two, at first sight apparently conflicting scenarios are in fact consistent: an ensemble of clumps with the given mass and size spectrum results in a fractal structure of the cloud image. This background suggests that the combined emission of an ensemble of clumps, where each clump is modeled as a spherical cloud in an embedding UV field, can reproduce the observed line intensities. In this paper we show, that the low-J CO and isotopomeric CO line emission ratios of spherical clumps are indeed rather insensitive to the details of the clump parameters (density, column density; and hence size and mass) and the strength of the embedding UV field (and hence position in the cloud). Further, we show that the line ratios match the narrow range of observed values. This then implies that the emission properties of the entire ensemble will be almost identical to those of the single clumps we study in this paper.

Several efforts have been made to model the observed 12CO/13CO line intensities emitted by higher density molecular condensations, exposed to FUV radiation in clumpy environments. Gierens et al. (1992) performed carbon monoxide line radiative transfer computations for spherically symmetric, FUV illuminated molecular clumps. The model reproduces successfully the low-J CO line intensities observed in the Orion A molecular cloud. Köster et al. (1994) presented the 12CO/13CO line intensities for finite-sized, plane-parallel PDRs.

In this paper we present a set of calculations over a large parameter range for the 12CO/13CO/C18O low-J line intensities emitted by spherical molecular clouds which are heated by an external FUV field. In contrast with the Gierens et al. models we do not use an analytical approximation for the temperature structure, and the CO and H2 density distribution, but we compute the PDR structure numerically and use the numerical results as input for the radiative transfer program of Gierens et al. This procedure provides the most realistic computation to-date of the low-J 12CO, 13CO and C18O emission line intensities which depend sensitively on both the CO density structures (for each isotope) and temperature distributions in the clouds. Unlike the plane-parallel models by Köster et al. we assume spherical geometry for the clouds (Störzer et al. 1996; see also Spaans & Neufeld 1997). As in Störzer et al. we consider clouds with gas densities which increase toward the cloud centers. In Sect. 2 we describe the basics of our model, in Sect. 3 we discuss the resulting 12CO/13CO/C18O low-J line intensities for a range of assumed hydrogen gas densities, column densities (i.e. cloud sizes), and FUV field intensities. In Sect. 4 we compare our results with observations, and summarize our results in Sect. 5.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000