Figure 6.4 Cloud geometry assumed for deriving the NH3 inversion spectrum.
The fact that the states along the rotational backbone are metastable means that we may effectively treat each one as an isolated two-level system. That is, we lump together all the levels on either side of the inversion transition into two fictitious "super-levels." Using such a model, one successful strategy has been to determine first the excitation temperature and optical thickness for both the (1,1) and (2,2) backbone states by matching theoretical and observed emission spectra. Next, one solves the rate equations governing the super-level populations within each backbone state, in order to relate the previously derived quantities to the cloud density and kinetic temperature.
Let us follow this procedure, covering the essential steps. For simplicity, we take the cloud to be a uniform-density sphere of radius R, and we consider a line of sight that passes through its center (Figure 6.4). We focus on the transfer of radiation through molecules that are in a specific backbone state, e.g., (1,1). Referring to the figure, the path length for radiation emanating from the cloud's far side is As = 2 R. The corresponding optical thickness is therefore Atv = 2 p kvR, where p and kv are the cloud's mass density and opacity, respectively. We now apply the detection equation (6.1), generalized to a frequency offline center:
In writing (6.15), we have used the Rayleigh-Jeans approximation to the function f, i. e., we have assumed Ta C Tex and Ta C Tbg. This approximation is, in fact, only marginally applicable because Ta for the inversion transition is about 1 K, while Tex and Tbg are between 3 and 20 K.
To generate theoretical spectra from equation (6.15), we must somehow account for the complex line structure. Within our two-level model, we let the opacity kv peak at discrete frequency intervals. Specifically, we represent Atv as a sum:
where the index i runs from 0 for the most intense central line to N = 17. Each line, assumed to be a Gaussian of identical width Av, is displaced in frequency by Av from the i = 0 line centered at va. The coefficients a are the relative absorption probabilities for each hyperfine transition. These probabilities are known from theory. Finally, At^ is the sum of the optical
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