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Limiting cases

The solution (44) readily illustrates the formation of spectral lines if we consider the variation of t (and also /0 and /s) with frequency. There are four cases to consider, one of which has two possibilities:

/0 = 0: there is no background radiation illuminating the cloud

/0 > 0 : background radiation illuminates the back of the cloud

(iii) t « 1: the gas is optically thin (for /s > /0 and /s < /0)

The condition /0 = 0 means that / (t ) will be affected only by radiation from the cloud. The t « 1 condition allows us to expand the exponential, e—T ~ 1—t . The solution (44) then reduces to

This tells us that the emission is proportional to the optical depth, for t « 1. This is reasonable because, for an observer located at t ~ 0 with leftward viewing detectors (Fig. 17), there are no atoms in view. The optical depth is zero and so is the detected intensity. As the observer moves to the right, toward increasing t , the number of atoms in the line of sight increases linearly with t . The cloud is optically thin so every layer dT of the cloud that is in view contributes equally to the intensity (Fig. 8.8); hence / a t. Note that changes in mass density p and opacity k along the line of sight are automatically incorporated into t .

Now we address the frequency variation of the quantities in (45). Let the atoms in the cloud have an atomic transition or resonance at some frequency. At that frequency the cross section a for absorption of incoming photons is high, and hence, so is the optical depth t. In general, t is a function of frequency and therefore so is the intensity /. We therefore rewrite (45) as

Resonances at two distinct frequencies are hypothesized and illustrated in Fig. 19a (left panel) which is a plot of t vs. v for an observer at fixed position x. From (46), we see that high optical depths at these frequencies lead to high emerging fluxes / (v) at these same frequencies, provided that /s is a smooth function of frequency. A plot of / vs. v for an arbitrarily chosen spectrum /s(v) (Fig. 19a, right panel) shows emission lines at the two resonance frequencies. Note that the spectrum lies well below the source spectrum /s because t « 1, in accord with (46).

Optical depth Specific intensity

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