For our sample observation, we use 7 = 0.3 to derive At012 = 27.
There are many variations on the basic method we have outlined. For example, an observer wishing to obtain higher angular resolution may prefer to use the J = 2 ^ 1 line of 12C16O at 1.3 mm instead of J =1 ^ 0. In this case, the optical depth will generally not be high enough for equation (6.3) to hold. Suppose, however, that the analogous line in 13C16O is still optically thin. Then one may observe another, even higher transition in 12C16O. One writes equation (6.1) for all three lines and solves for the unknowns Tex, At013, and At012. These optical depths refer to the J =2 ^ 1 transition only, since the higher one can be obtained through the assumption of LTE. We will encounter a practical application of this technique in Chapter 13, when discussing the temperatures within molecular outflows.
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