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Figure 11.18. Plots of intensity I(t) vs. optical depth t from (44) for two cases: (a) source (cloud) intensity greater than the background intensity, Is > I0, and (b) Is < I0. As frequency is varied, the optical depth becomes higher at a resonance. If depth "A" is off resonance and depth "B" is centered at the resonance, case (a) yields an emission line and case (b) an absorption line.

Figure 11.18. Plots of intensity I(t) vs. optical depth t from (44) for two cases: (a) source (cloud) intensity greater than the background intensity, Is > I0, and (b) Is < I0. As frequency is varied, the optical depth becomes higher at a resonance. If depth "A" is off resonance and depth "B" is centered at the resonance, case (a) yields an emission line and case (b) an absorption line.

The RTE (39) can be solved for I (t) by integration as follows. Multiply (39) by eT, eT + I eT = Is eT, at rewrite the left side as d(IeT)/dT, and integrate from 0 to t, f d(I eT) = f Is eT dT Jo Jo

For our cloud with Is independent of optical depth t ,

Insert the limits and divide through by eT,

(Solution of radiative transfer equation)

This is the solution of the RTE. The first term on the right shows the decreasing effect of the background radiation I0 as the optical depth increases, while the second term shows the increasing effect of the source (cloud) emission. These two terms and their sum are plotted in Fig. 18. These plots illustrate the variation of intensity with t for a single chosen frequency.

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