At fixed Tg and Ay, equation (7.34) shows that FCo( J + 1,J) is a function only of the variable a = ( J +1)/0. The cooling flux initially rises as a3, peaks at a « 4, and thereafter declines. It would seem, then, that the peak flux might emanate from that transition with J « 40 — 1. In practice, however, this line is always optically thin, with associated level populations so far below LTE that it contributes little to the net cooling. The true maximum in emission occurs at the lower value J* given implicitly by equation (7.32) and depicted in Figure 7.9. Because of the steep rise in emission with a (or J*), much of the total CO cooling arises from this single, critical line.
To obtain the luminosity from the J* + 1 ^ J* transition, we multiply the flux in equation (7.34) by the cloud surface area. Alternatively, since the line is only marginally optically thick, we may calculate a volumetric cooling rate ACO. This form is more convenient when comparing the rate to that from other sources. For the volumetric form, we may employ our previous results for two-level atoms. Assuming that radiative trapping can maintain the J* + 1 level in LTE, we use equation (7.25) to find
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