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Figure 7.9 Plot of Jt, the highest CO rotational level creating an optically thick line, shown as a function of gas temperature. Also plotted is the volumetric cooling rate A£O. Here we have displayed Jt as a smoothly varying real number, to avoid artificial jumps in the associated cooling rate.

At fixed 0, the presence of the Boltzmann factor exp[— J (J + I)/20] ensures that TJ+1jJ falls to unity at a modest J -value, which we denote as J*. We may find this critical level by solving exp

ZAwNcoc3 ( J*+ l-exp[-(J« + l)/0] 16 7r z/?n AV I 2 J* + 1) 0

In the second form of this equation, we have inserted the expression for t10 from (7.28). Figure 7.9 shows J* as a function of temperature, computed for a cloud with AV = 1 km s-1, nH2 = 1 x 103 cm-3, and a diameter of 1 pc, i. e., with an associated hydrogen column density of Nh = 6 x 1021 cm-2. In this example, NCO =2 x 1018 cm-2. The rise in J* with temperature demonstrates the increasing importance of radiative trapping.

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