6.1.5 The X-Factor
As one considers more distant clouds within the Milky Way, or those in other galaxies, the emission from 13C16O becomes too feeble for practical use, even if equation (6.13) still applies in principle. Observational surveys on the largest scales revert to 12C16O and utilize only the velocity-integrated profile. If we again assume beam dilution and efficiency factors of unity, the hydrogen column density follows from the simple empirical relation:
The proportionality constant X is currently estimated at 2 x 1020 cm-2 K-1 km-1 s, again with a 50 percent uncertainty in either direction. Satellite observations of 7-rays have played a major role both in establishing equation (6.14) and in setting the value of X. We will see in Chapter 7 that cosmic-ray protons penetrating molecular clouds produce these high-energy photons through collisions with hydrogen nuclei. With sufficient knowledge of the cosmic-ray flux, the 7-radiation provides a direct measure of NH, to be compared with 12C16O observations of the same region.
What is the theoretical basis for equation (6.14)? Earlier, we emphasized that the optical thickness of 12C16O implies that its integrated intensity should not be proportional to the column density along any line of sight. However, this statement refers to a well-resolved cloud region. The emission at great distances actually stems from an ensemble of many clouds. For equation (6.14) to hold, this diffuse collection must radiate, in some sense, as if it were optically thin, even if each individual cloud is not.
This viewpoint presupposes a certain degree of uniformity among all the radiating entities, at least in a statistical sense. To elucidate the underlying assumptions, let us crudely estimate the integral in equation (6.14) as T^ AVr, where T^ is the antenna temperature at line center from a single, unresolved cloud, and AVr is the cloud's observed line width. Equations (6.3) and (6.4) together imply that T^ depends only on the local gas kinetic temperature. From the discussion in § 3.3, we may relate AVr to the virial velocity in equation (3.20). This velocity,
in turn, is proportional to nH L. Here, nH and L represent mean values within the smallest radiating unit, which, for large-scale surveys, could be an entire cloud complex. We conclude that the "constant" X in equation (6.14) is actually proportional to n1I/2/T^. It is hardly surprising, then, that careful studies have indicated variations in X across our own and external galaxies, with a significant decline close to the galactic centers. Nevertheless, the figure quoted above serves as a reasonable first approximation in many circumstances.1
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