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Figure 5.7 Emission in the J = 1 ->■ 0 line of 12C16O. The CO is immersed in a gas of pure H2. Emission is measured per CO molecule and is displayed as a function of H2 number density.

where g1 and g0 are the degeneracies of the two levels. In the case of CO rotational states, gJ = 2 J +1. For ntot C ncrit, n1/n0 is small and proportional to ntot. (See Appendix B.) The excitation temperature in this case is less than Tkin, the kinetic temperature that characterizes the velocity distribution of the colliding molecules. For ntot > ncrit, however, the CO molecule comes into local thermodynamic equilibrium (LTE) with its environment. The J =1 and J = 0 level populations are again related through equation (5.14), but with Tex now equal to Tkin.

Increasing the density in a cloud can therefore enhance the J =1 ^ 0 emission, but only for subcritical values of ntot. We show the full behavior in Figure 5.7, which was obtained by numerical calculation of the level populations. At a fixed value of Tkin, the 1 ^ 0 emission rate peaks as ntot increases to ncrit and beyond. The high-density decline is caused by the increasing excitation of molecules to the J > 1 states. This effect slowly drains the population of the J =1 level, eventually forcing it down to the LTE value. The calculation here ignores the fact that many of the emitted 1 ^ 0 photons can themselves excite CO molecules instead of leaving the cloud. Inclusion of this radiative trapping would cause the peak in emission to be achieved at densities somewhat less than ncrit.

Figure 5.7 concerns a parcel of gas with uniform density ntot. Along any line of sight through a molecular cloud, the true density will vary, with most material being at some mini law:

Figure 5.7 Emission in the J = 1 ->■ 0 line of 12C16O. The CO is immersed in a gas of pure H2. Emission is measured per CO molecule and is displayed as a function of H2 number density.

mum "background" value. If this ambient gas has an ntot well below ncrit for the transition of interest, it will not contribute appreciably to the emission. On the other hand, we have seen that the emissivity declines for ntot far above ncrit, where there is often little material in any case. The conclusion, valid beyond the specific case of CO, is that observations in a given transition are most sensitive to gas with densities near the corresponding ncrit. This fact should be borne in mind when interpreting molecular line studies.

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