2. Spherical PDR models
Our method for calculating the chemical and physical structure of the spherical PDRs is described in detail in Störzer et al. (1996). The basic assumptions of our models are as follows. First, the clouds are assumed to be spherically symmetric. Second, the clouds are exposed to isotropic FUV fields. Third, the total hydrogen particle density increases radially inward and is described by a power law of the form , where is the density at the clump surface and R is the total clump radius.
The main parameters of the models are: the volume average hydrogen particle density , the average projected hydrogen particle column density and the intensity of the FUV field relative to the mean interstellar UV field (Draine 1978). The models presented here differ in three ways from the models presented in Störzer et al. (1996). First, we use the photoelectric heating rates of Bakes & Tielens (1994). The Bakes & Tielens (1994) photoelectric heating rates are larger in the CO emitting regions than the previously used rates of Draine (1978), resulting in somewhat stronger CO line intensities.
Second, we have included the 18O chemistry. We assume a [12C]/[13C] abundance ratio of 67 and a [16O]/[18O] abundance ratio of 500. The chemical network includes the isotopic fractionation reactions listed by Langer et al. (1984). It is based on the most recent release of the UMIST network (Millar et al. 1997) with the relevant isotopically substituted C and O bearing species and corresponding reaction rates added, similar to what is described in Le Bourlot et al. (1993). We use an unattenuated CO photodissociation rate of and employ the 12CO/13CO/C18O self-shielding factors given by van Dishoeck & Black (1988).
Third, the dust temperature is calculated by using the formalism described in Hollenbach et al. (1991). This formalism takes into account that the dust inside the clouds is heated by a combination of the externally incident FUV radiation and, at large cloud depth, by the infrared (IR) continuum radiation emitted by warm dust in the FUV heated surface layers. Heating by IR radiation results in dust temperatures of 10 - 40 K in the cloud cores, depending on the strength of the incident FUV field. The warm dust in the FUV shielded cores is particularly important for the C18O lines which are produced at large cloud depths.
As is discussed in Störzer et al. (1996), in our spherical PDR model we treat the line radiative transfer using an escape probability formalism when iteratively calculating the thermal structure as a balance between the gas heating and cooling. We use a more sophisticated multi-shell radiative transfer code to calculated the emergent line intensities after the final thermal and chemical structure of the spherical PDR is established. For those species considered in the heating and cooling balance, the differences between the line intensities that would be derived in the escape probability formalism and those calculated in the full radiative transfer are marginal.
In order to achieve a reasonably fast numerical treatment of each spherical PDR model, we had to keep the chemistry relatively simple. In particular, we did not including sulphur bearing species. Sulphur may affect the degree of ionization in the cloud, and has some effect on the location and temperature of the CI/CO transition layer (see e.g. Sternberg & Dalgarno 1995). Proper treatment of sulphur within the chemical network should thus be included in future applications of our spherical PDR model, but was beyond the scope of the present study.
In the following models we concentrate on FUV fields in the range , which are typical values for many star forming cloud complexes, and vary the two other parameters between and . This range of densities and column densities corresponds to cloud sizes of pc to pc. Other model parameters are listed in Table 1.
Table 1. Model parameters
The 12CO/13CO/C18O low-J line center brightness temperatures (see Köster et al. 1994 for a definition), i.e. the intensities of the , and transitions, presented in the following section are given by the projected average of the intensity over the clump surface (see Störzer et al. 1996). For large clouds , where the PDR is a thin surface shell, corresponds very well to the intensity emitted by a plane-parallel, semi-infinite slab. However, for small clouds, where most of the carbon is in the form of C+ and CO is only formed in the high density core, can differ significantly from values produced in plane-parallel models. Limb-brightening effects have a less important effect on the emitted line intensities.
© European Southern Observatory (ESO) 2000
Online publication: June 8, 2000