Absorption of radiation. Radiation enters from the left.Any beam striking a sphere is absorbed.
We define the optical depth to be this quantity: t = nal (6.7)
Our requirement in equation (6.4) reduces to t V 1. Under this restriction, the optical depth of any section of material is simply the fraction of incoming radiation that is absorbed when the radiation passes through that material. (For example, if the optical depth is 0.01, then 1% of the incoming radiation is absorbed.)
In general, a will be a function of wavelength. For example, we know that at a wavelength corresponding to a spectral line, a particular atom will have a very large cross section for absorption. At a wavelength not corresponding to a spectral line, the cross section will be very small. To remind us that a is a function of A (or v), we write it as aA (or av). This means that the optical depth is also a function of A (or v), so we rewrite equation (6.7) as
In terms of these quantities, the optical depth is given by tx = nlo-x
In the above discussion, we required that the optical depth be much less than unity. Our interpretation of t as the fraction of radiation absorbed only holds for t V1. What if that is not the case? We then have to divide our cylinder into several layers. If we make the layers thin enough, we can be assured that the optical depth for each layer will be very small. We then follow the radiation through, layer by layer, looking at the fraction absorbed in each layer (Fig. 6.4).
In our discussions, the quantity nl occurs often. It is the product of a number density and a length, so its units are measured in number per unit area. It is the number of particles along the full length, l, of the cylinder per unit surface area. For example, if we are measuring lengths in centimeters, it is the number of particles in a column whose face surface area is 1 cm2, and whose length is l, the full length of the cylinder. We call this quantity the column density.
We can see that the optical depth depends on the properties of the material - e.g. cross section and density of particles - and on the overall size of the absorbing region. It is sometimes convenient to separate these two dependencies by defining the absorption coefficient, which is the optical depth per unit length through the material,
If ka gives the number of absorptions per unit length, then its inverse gives the mean distance between absorptions. This quantity is called the mean free path, and is given by
Let's look at the radiation passing through some layer with optical depth dr. Since dr V 1, it is the fraction of this radiation that is absorbed. The amount of radiation absorbed in this layer is I dr. The amount of radiation passing through to the next layer is I(1 — dr). The change in intensity, dl, while passing through the layer, is dI = Io
Notice that dI is negative, since the intensity is decreased in passing through the layer.
Now that we know how to treat each layer, we must add up the effect of all the layers to find the effect on the whole sample of material. We can see that we have formulated the problem so that we are following I as a function of r. We let r' be the optical depth through which the radiation has passed by the time it reaches a particular layer, and I' be the intensity reaching that layer. Then r' ranges from zero, at the point where the radiation enters the material, to r, the full optical depth where the radiation leaves the material. Over that range, I' varies from I0, the incident intensity, to I, the final intensity. Using equation (6.14),
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