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

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3. Line intensities

3.1. Fixed FUV field intensity

Fig. 1 displays contours of constant 12CO 2[FORMULA]1 line center brightness temperatures, [FORMULA], as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for a fixed value of the FUV field intensity, [FORMULA]. For fixed [FORMULA] the line intensity increases with [FORMULA] because the chemical formation of CO is more efficient for larger densities compared to the CO photodestruction. The CO emission region is, therefore, shifted in layers with higher temperatures. For small clouds, i.e. small values for [FORMULA], CO is present only in the clump cores, so that [FORMULA] decreases for fixed [FORMULA] and decreasing column densities [FORMULA]. The 12CO 2[FORMULA]1 line center brightness temperature varies from 1.1 K for a model with [FORMULA] and [FORMULA] to 53 K for a model with [FORMULA] and [FORMULA]. The corresponding 13CO 2[FORMULA]1 line intensities vary from 0.5 K to 24 K. C18O is nearly fully dissociated in small clumps, and the C18O 2[FORMULA]1 intensities reach maximum values of about 4 K in the limit of large [FORMULA] and [FORMULA].

[FIGURE] Fig. 1. Contours of the 12CO 2[FORMULA]1 line center brightness temperature (in K) as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for [FORMULA].

Fig. 2 displays the 12CO 2[FORMULA]1/12CO 1[FORMULA]0 line center brightness temperature ratio in the [FORMULA] vs. [FORMULA] plane. This line ratio is insensitive to the density or cloud size. For large [FORMULA] and densities smaller than about [FORMULA] the line ratio is less than 1. At higher densities the ratio is slightly larger than 1. When the column density [FORMULA] is large the 12CO lines are emitted from the surface layers of the spherical clumps where the gas densities are equal to about [FORMULA]/2. The critical density for collisional de-excitation of the 1[FORMULA]0 transition is [FORMULA] and the [FORMULA] level population is always thermalized in our models. However, the critical density for the 2[FORMULA]1 transition is [FORMULA] so that in our lower density models the [FORMULA] levels are subthermally excited. Therefore, the 12CO 2[FORMULA]1/12CO 1[FORMULA]0 line ratio is less than 1 in the low density, large column density models. At high densities the [FORMULA] level is thermalized as well, but because of the higher optical depth of the 2[FORMULA]1 transition it is emitted from warmer layers than the 1[FORMULA]0 line, and the 12CO 2[FORMULA]1/12CO 1[FORMULA]0 ratio is larger than 1. When [FORMULA] and [FORMULA] are both small the CO lines are formed mainly in the cloud cores. Because of the density gradient the cores are denser than the surface layers and therefore the line ratio increases with [FORMULA] when [FORMULA] is small ([FORMULA]). For higher densities the 2[FORMULA]1/1[FORMULA]0 ratio decreases with [FORMULA] because limb-brightening becomes more important for smaller sized clouds.

[FIGURE] Fig. 2. Contours of the 12CO 2[FORMULA]1/12CO 1[FORMULA]0 line center brightness temperature ratio as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for [FORMULA].

Fig. 3 displays the 13CO 2[FORMULA]1/13CO 1[FORMULA]0 line ratio. The behavior is similar to the 12CO 2[FORMULA]1/12CO 1[FORMULA]0 ratio. For fixed [FORMULA] the ratio decreases for increasing [FORMULA], and for fixed [FORMULA] the ratio increases with [FORMULA]. Because of the smaller optical depths of the 13CO lines compared with the 12CO lines the 13CO line ratio attains somewhat larger values.

[FIGURE] Fig. 3. Contours of the 13CO 2[FORMULA]1/13CO 1[FORMULA]0 line center brightness temperature ratio as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for [FORMULA].

The 12CO 3[FORMULA]2/2[FORMULA]1 and 13CO 3[FORMULA]2/2[FORMULA]1 ratios displayed in Figs. 4 and 5 show a similar behavior. For gas densities less than the critical density ([FORMULA]) of the 3[FORMULA]2 transition the ratios are smaller than 1, and increase above unity at larger densities. However, the variation in the line ratio is even smaller than for the 2[FORMULA]1/1[FORMULA]0 ratios.

[FIGURE] Fig. 4. Contours of the 12CO 3[FORMULA]2/12CO 2[FORMULA]1 line center brightness temperature ratio as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for [FORMULA].

[FIGURE] Fig. 5. Contours of the 13CO 3[FORMULA]2/13CO 2[FORMULA]1 line center brightness temperature ratio as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for [FORMULA].

The isotopic ratio of the 12CO and the 13CO 2[FORMULA]1 lines is displayed in Fig. 6. The ratio ranges from 1.8 when [FORMULA] is small and [FORMULA] is large, to values of 8.0 when [FORMULA] is large and [FORMULA] is small. This ratio is sensitive to the temperature gradient. The 12CO lines are formed in warm FUV heated surface layers, whereas the 13CO lines are emitted from cooler regions closer to the cloud cores. This line ratio is also affected by the different optical depths of the 12CO and the 13CO 2[FORMULA]1 lines. This difference implies that the spherical clouds are effectively smaller for the 13CO transitions than for the 12CO transitions. The effect of the temperature gradient is most important for the high-density clouds, and the optical depth effect becomes significant for low [FORMULA] (i.e. small) clouds. The line ratio therefore increases as [FORMULA] increases and [FORMULA] decreases.

[FIGURE] Fig. 6. Contours of the 12CO 2[FORMULA]1/13CO 2[FORMULA]1 line center brightness temperature ratio as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for [FORMULA].

Fig. 7 shows the isotopic ratio of the 12CO and the C18O 2[FORMULA]1 line intensities. The behavior is similar to the 12CO/13CO ratio, but with larger values due to the much smaller C18O abundances. The 12CO/C18O 2[FORMULA]1 ratio varies from 3 to [FORMULA], reaching the large values at low column densities and high densities where the total amount of C18O in the clump gets small enough that the C18O lines actually become optically thin.

[FIGURE] Fig. 7. Contours of the 12CO 2[FORMULA]1/C18O 2[FORMULA]1 line center brightness temperature ratio as a function of the volume average hydrogen particle density [FORMULA] and the average projected hydrogen particle column density [FORMULA] for [FORMULA].

3.2. Fixed density

Figs. 8-10 display the 12CO 2[FORMULA]1 line intensities, and the isotopic 12CO/13CO and 12CO/C18O 2[FORMULA]1 line ratios as functions of the FUV intensity [FORMULA] and the cloud column density [FORMULA]. In these models the hydrogen particle density was kept fixed at [FORMULA]=[FORMULA]. For [FORMULA] the line intensities and ratios are insensitive to [FORMULA]. The 12CO 2[FORMULA]1 line intensities increase for increasing cloud size. Stronger FUV fields result in somewhat weaker line intensities, especially for small clouds. The isotopic ratios are large when [FORMULA] and [FORMULA] are large, and relatively small for large clumps and weak FUV fields. The line ratios are mainly affected by the different optical depths of the 12CO, 13CO and C18O 2[FORMULA]1 lines. This difference implies that the spherical clouds are effectively smaller for the C18O and 13CO transitions than for the 12CO transitions. Temperature gradient effects are less important.

[FIGURE] Fig. 8. Contours of the 12CO 2[FORMULA]1 line center brightness temperature (in K) as a function of the average projected hydrogen particle column density [FORMULA] and the FUV field [FORMULA] for a volume average hydrogen particle density [FORMULA]=[FORMULA].

[FIGURE] Fig. 9. Contours of the 12CO 2[FORMULA]1/13CO 2[FORMULA]1 line center brightness temperature ratio as a function of the average projected hydrogen particle column density [FORMULA] and the FUV field [FORMULA] for a volume average hydrogen particle density [FORMULA]=[FORMULA].

[FIGURE] Fig. 10. Contours of the 12CO 2[FORMULA]1/12C18O 2[FORMULA]1 line center brightness temperature ratio as a function of the average projected hydrogen particle column density [FORMULA] and the FUV field [FORMULA] for a volume average hydrogen particle density [FORMULA]=[FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000
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