Although a number of molecules, such as H2O and O2, act as cooling agents, the dominant one is CO. Here, the J =1 ^ 0 rotational transition is always optically thick in molecular clouds. The relevant cross sections are so large, in fact, that a number of higher transitions are also optically thick. Internal trapping reduces the emission in any given line, while enhancing the populations of the higher levels above those attainable through collisions alone.
Consider now an idealized cloud of spatially uniform density and temperature. What is the optical depth at line center for the J =1 ^ 0 transition? By using equation (6.9), now applied to the main isotope 12C16O, we may write t10 in terms of the column density NCO:
A10 Nco c3 8tt z/f0 Q Ay
hv io ksTg
Here we have denoted the line-center frequency by v10 and have set Tex equal to Tg. As we remarked in § 6.2, this LTE assumption is justified once the line radiation becomes optically thick. In writing (7.28), we have also assumed that the broadening of the spectral line is mainly due to bulk motion in the cloud interior. Thus, we have set the line width Av10 equal to v10 AV/c, where AV is the internal velocity dispersion. For the partition function Q, we have used equation (C.18) to write Q = 2kBTg/hv10. After expanding the exponential, whose argument is less than unity, we may evaluate t10 numerically as t10 = 8 x 102
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