e-T vs. t, showing the fall-off in transmitted radiation as the optical depth increases. Note that the curve looks almost linear for small t. For large t, it approaches zero asymptotically.

ex = 1 + x, for x V 1. In this case, equation (6.18) becomes

In this form, all of the I' dependence is on the left, and the r' dependence is on the right. To add up the effect of the layers, we integrate equation (6.15) between the limits given above:

Raising e to the value on each side, remembering that eln x = x, and multiplying both sides by I0 gives

We can check this result in the limit r V 1, called the optically thin limit, using the fact that

This is the expected result for small optical depths, where r again becomes the fraction of radiation absorbed.

As shown in Fig. 6.5, e—r falls off very quickly with r. This means that to escape from the Sun, radiation must come from within approximately one optical depth of the surface. This explains why we only see the outermost layers. Since the absorption coefficient is a function of wavelength, we can see to different depths at different wavelengths. At a wavelength where kx is large, we don't see very far into the material. At wavelengths where kx is small, it takes a lot of material to make rx = 1. We take advantage of this to study conditions at different depths below the surface.

So far we have only looked at the absorption of radiation passing through each layer. However, radiation can also be emitted in each layer, and the amount of emission also depends on the optical depth. In general, we must carry out complicated radiative transfer calculations to take all effects into account. To solve these problems, we use powerful computers to make mathematical models of stellar atmospheres. In these

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