According to equation (6.22), Trad = PTbg + (1 — P)Tex in the same limit. After substituting these results into equation (B.10) and expanding the exponentials, we find
We recall that ncrit = Au\/yu1, so that the dimensionless quantity ntot/pncrit in (6.25) measures the relative rates of collisional and radiative deexcitation. The factor P modifies the net emission rate to account for the sphere's finite optical thickness. It is instructive to rewrite equation (6.24) in the form
We see that Tex is a weighted average of Tbg and Tkin. Under conditions such that
P ncrit To we have a C 1 and LTE applies: Tex « Tkin. In an optically-thin environment, i.e., when p < 1, this condition is fulfilled once the density of colliders ntot exceeds ncrit by a sufficient amount. Conversely, when collisions are relatively rare, a « 1 and the two-level system comes into thermal equilibrium with the background radiation field: Tex « Tbg. Notice, finally, that a can be much less than unity at any density ntot, provided p is sufficiently small. In other words, large optical thickness drives the level populations into LTE, even if the ambient density is subcritical. This effect of radiative trapping, illustrated here for the specific case of a two-level system, is of general importance in molecular clouds.
Suppose we know both Tex and Ar0ot for the (1,1) system through matching to the observed hyperfine spectrum. With 4>(v) given by equation (6.20), we may calculate the escape probability p from (6.23), after finding At0 from (6.17). We then substitute the values of p and Tex into equation (6.24) to yield a relation between ntot and Tkin. If we now repeat the procedure for the (2,2) system, we obtain a second such relationship, from which the density and kinetic temperature follow. For the representative dense core in Figure 6.5, this method yields ntot = 2.5 x 104 cm~3 and Tkin = 10 K.
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