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

3. Line intensities

3.1. Fixed FUV field intensity

Fig. 1 displays contours of constant ¹²CO 21 line center brightness temperatures, , as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for a fixed value of the FUV field intensity, . For fixed the line intensity increases with 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 , CO is present only in the clump cores, so that decreases for fixed and decreasing column densities . The ¹²CO 21 line center brightness temperature varies from 1.1 K for a model with and to 53 K for a model with and . The corresponding ¹³CO 21 line intensities vary from 0.5 K to 24 K. C¹⁸O is nearly fully dissociated in small clumps, and the C¹⁸O 21 intensities reach maximum values of about 4 K in the limit of large and .

Fig. 1. Contours of the 12CO 21 line center brightness temperature (in K) as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for .

Fig. 2 displays the ¹²CO 21/¹²CO 10 line center brightness temperature ratio in the vs. plane. This line ratio is insensitive to the density or cloud size. For large and densities smaller than about the line ratio is less than 1. At higher densities the ratio is slightly larger than 1. When the column density is large the ¹²CO lines are emitted from the surface layers of the spherical clumps where the gas densities are equal to about /2. The critical density for collisional de-excitation of the 10 transition is and the level population is always thermalized in our models. However, the critical density for the 21 transition is so that in our lower density models the levels are subthermally excited. Therefore, the ¹²CO 21/¹²CO 10 line ratio is less than 1 in the low density, large column density models. At high densities the level is thermalized as well, but because of the higher optical depth of the 21 transition it is emitted from warmer layers than the 10 line, and the ¹²CO 21/¹²CO 10 ratio is larger than 1. When and 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 when is small (). For higher densities the 21/10 ratio decreases with because limb-brightening becomes more important for smaller sized clouds.

Fig. 2. Contours of the 12CO 21/12CO 10 line center brightness temperature ratio as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for .

Fig. 3 displays the ¹³CO 21/¹³CO 10 line ratio. The behavior is similar to the ¹²CO 21/¹²CO 10 ratio. For fixed the ratio decreases for increasing , and for fixed the ratio increases with . Because of the smaller optical depths of the ¹³CO lines compared with the ¹²CO lines the ¹³CO line ratio attains somewhat larger values.

Fig. 3. Contours of the 13CO 21/13CO 10 line center brightness temperature ratio as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for .

The ¹²CO 32/21 and ¹³CO 32/21 ratios displayed in Figs. 4 and 5 show a similar behavior. For gas densities less than the critical density () of the 32 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 21/10 ratios.

Fig. 4. Contours of the 12CO 32/12CO 21 line center brightness temperature ratio as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for .

Fig. 5. Contours of the 13CO 32/13CO 21 line center brightness temperature ratio as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for .

The isotopic ratio of the ¹²CO and the ¹³CO 21 lines is displayed in Fig. 6. The ratio ranges from 1.8 when is small and is large, to values of 8.0 when is large and 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 21 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 (i.e. small) clouds. The line ratio therefore increases as increases and decreases.

Fig. 6. Contours of the 12CO 21/13CO 21 line center brightness temperature ratio as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for .

Fig. 7 shows the isotopic ratio of the ¹²CO and the C¹⁸O 21 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 21 ratio varies from 3 to , 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.

Fig. 7. Contours of the 12CO 21/C18O 21 line center brightness temperature ratio as a function of the volume average hydrogen particle density and the average projected hydrogen particle column density for .

3.2. Fixed density

Figs. 8-10 display the ¹²CO 21 line intensities, and the isotopic ¹²CO/¹³CO and ¹²CO/C¹⁸O 21 line ratios as functions of the FUV intensity and the cloud column density . In these models the hydrogen particle density was kept fixed at =. For the line intensities and ratios are insensitive to . The ¹²CO 21 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 and 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 21 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.

Fig. 8. Contours of the 12CO 21 line center brightness temperature (in K) as a function of the average projected hydrogen particle column density and the FUV field for a volume average hydrogen particle density =.

Fig. 9. Contours of the 12CO 21/13CO 21 line center brightness temperature ratio as a function of the average projected hydrogen particle column density and the FUV field for a volume average hydrogen particle density =.

Fig. 10. Contours of the 12CO 21/12C18O 21 line center brightness temperature ratio as a function of the average projected hydrogen particle column density and the FUV field for a volume average hydrogen particle density =.

© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000
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