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At resonance frequencies, with their higher cross sections, the gas behaves as if it contains more atoms, or as if it were thicker. In this case, the observer would seem to "see" more emitting atoms, and hence greater intensity. At frequencies adjacent to a line, t will be lower by definition and fewer atoms are seen. If t is constant at these adjacent frequencies, as in the left panel of Fig. 19a, the output spectrum I(v) will, according to (46), mimic the spectrum Is away from the line as shown in the right panel.

Case 2: I0 = 0, t» 1 Here the gas is very thick (t » 1), and the expression (44) reduces to,

which we have previously deduced (40); here we use the variable v rather than t(v).

The output radiation given by (47) equals that of the continuum source specific intensity at all frequencies. If the source spectrum is the blackbody spectrum, the output spectrum at any depth t » 1 is also blackbody. Even though the resonances may exist (Fig. 19b), the intensity I(v) has no dependence on t, and hence no spectral lines will form.

In this case, the local observer sees the maximum possible number of emitting atoms at any frequency so the resonances are not apparent. It is like being immersed in a thick fog that is denser (more opaque) in some directions than others. Nevertheless, the appearance in all directions (frequencies) is uniform as long as the fog is totally impenetrable (t » 1) in all directions.

The observer in the fog sees only to a depth that yields enough water droplets to completely block the view. The same number of water droplets are thus seen in all directions, and the view appears uniform even though in some directions it penetrates less deeply than others. In our case, the increase of opacity at some frequency reduces the depth of view, but the observed intensity does not change.

The optically thick character of the gas allows the photons and particles to interact sufficiently to come into equilibrium thus giving rise to the continuum (blackbody) spectrum characteristic of the cloud.

Case 3: I0 > 0, t < 1 In this case, there is a source behind the cloud. Since t < 1, we again use the Taylor expansion, e—T ~ 1 — t , so that (44) becomes

I = I0 + t(Is — I0) (I0 > 0, t < 1) (11.48)

Consider two cases here, Is > I0 and Is < I0. In the former case, the output intensity is the background intensity I0 plus another positive term. If the optical depth t is higher at some frequency (i.e., greater opacity k) than at surrounding frequencies, the emerging flux will be greater at that frequency. This yields an emission line (Fig. 19c).

In the case of Is < I0, the rightmost term is negative, and the emerging intensity is less than the background intensity. If again, the optical depth t is especially large at some frequency, the emerging intensity is depressed even more at that frequency. This yields an absorption line (Fig. 19d).

These same conclusions extend to somewhat larger optical depths, t < 2, as illustrated in the plots of the function I (t) vs. t (Fig. 18). In the case of Is > I0 (Fig. 18a), the observed intensity I increases with optical depth t. At a given frequency not at a resonance, the intensity I might be given by the value at point A in the plot. At the frequency of a resonance, the optical depth is higher (by definition), and the observed intensity is therefore higher (point B). Thus an emission line is observed. For Is < I0 (Fig. 18b), the increase in opacity again moves the observer from A to B, but in this case it yields a decrease in intensity, or an absorption line.

If the functions I0 and Is are each blackbody spectra, the one with the higher temperature will have the greater intensity at any frequency (Fig. 8). Thus we have Ts > T0 for the Is > I0 case, and Ts < T0 for the Is < I0 case. We conclude therefore that if the temperature of the foreground cloud Ts is greater than the background temperature T0, a spectrum with emission lines will emerge, and that if the cloud is cooler than the background, a spectrum with absorption lines will emerge.

In most stellar atmospheres at the depth seen in visible light (the photosphere), the temperature decreases with altitude, i.e., toward the observer. The deeper hot layers are then the background radiation for the higher, cooler regions. Absorption lines are thus prevalent in stellar spectra at visible wavelengths.

In contrast, radiation from the sun at ultraviolet frequencies yields emission lines. The observed ultraviolet radiation comes from high in the solar atmosphere because the higher opacities in the ultraviolet limit the depth into which the observer can "see". In these higher chromospheric regions, the temperature is increasing with height (moving toward the 106-K corona). Thus the higher temperatures are in the foreground, and the spectra characteristically exhibit emission lines.

Case 4: I0 > 0, t » 1 In this case, the gas is optically thick and (44) again yields

This is the same expression obtained when there was no background intensity (47). Since the gas is optically thick, the presence of the background source is immaterial (Fig. 19e). The radiation at any t is simply the continuum source (blackbody) spectrum of the optically thick cloud. It does not matter whether I0 > Is or I0 < Is.

Summary

This concludes our discussion of the limiting cases of the solution to the radiative transfer equation. In each case the result is a continuum spectrum that reflects one or both of the spectra /0 and /s with, in some cases, superimposed lines created by increases in the optical depth t at certain frequencies. If the foreground cloud intensity (temperature) is greater than the background intensity (temperature), emission lines are formed. If the opposite is true, absorption lines are formed.

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