number counts of background stars number counts of background stars
13c160 /= 1 emission line
Figure 6.2 Steps in deriving the relationship between the 13C16O and the total hydrogen column densities.
Here Av13 is the observed full-width half-maximum of the J =1 ^ 0 line. The derivation of equation (6.9) assumes this width is intrinsic and neglects any saturation broadening. The quantity Q13 is the partition function for the rotational levels. Equation (C.18) implies that Q13 = 2 Te13/T013. Returning once more to our example, the observed velocity width, AVr13 = (Av 13/v0) c, is 1.5 km s"1. Having already determined Ar013, we deduce that NCO = 8.8 x 1015 cm"2.
For a given abundance of CO relative to hydrogen, NCO should be proportional to the total hydrogen column density NH = NHi + 2 NH2. Knowing NH will not give us, of course, the local volume density at interior points. However, if the cloud is well mapped, we may be able to estimate the physical depth of the cloud along our line of sight at a number of points. Dividing the column density by this depth then gives the average hydrogen number density nH in the appropriate column.
How, then, do we obtain the hydrogen column density from the CO data? The usual procedure is to invoke an empirical NH - N1O relationship. As this relationship, or analogous ones involving other CO isotopes, underlies the mass and density estimates for many molecular clouds and cloud complexes, we should understand its derivation. The essential steps, which rely on observations from ultraviolet to millimeter wavelengths, are shown schematically in Figure 6.2.
We recognize at the outset that the desired relationship cannot be established simply from measurements of quiescent molecular clouds, where it is impossible to observe NH directly. We saw in Chapter 2, however, that O and B stars located behind diffuse clouds can yield estimates of NHI through photo-excitation of the Lya transition. A similar procedure, but employing Lyman band excitation, may be used to obtain H2 column densities in front of early-type stars. In this manner, we may estimate the total NH for clouds that have both atomic and molecular hydrogen. Unfortunately, the clouds for which the procedure works are not detectable in 13C16O. What we must do is relate the column density to some other parameter that can be observed in both diffuse and molecular clouds. Such a parameter is AV. This is obtained from the color excesses of the same background O and B stars used to determine hydrogen column densities. In fact, stellar absorption dips and color excesses were used to establish the linear NH — EB-V relation of equation (2.46). We then used equation (2.16) to establish the proportionality of Nh and AV in equation (3.2). Although derived from observations of diffuse clouds, this latter equation can safely be extended to molecular clouds, as long as the composition of embedded dust grains is not significantly different.
The final step is to establish a connection between NCO and AV. Returning to molecular clouds, we have already seen how the first quantity can be obtained from the measured intensity of the J =1 ^ 0 emission line. If a cloud is not too opaque, we may estimate AV by considering the obscuration of background stars. Consider for simplicity a uniform distribution of identical field stars, with spatial density n* and a single absolute visual magnitude MV. Suppose further that a molecular cloud, subtending a solid angle Qc, exists between distances r1 and r2, and provides the only source of visual extinction (see Figure 6.3). Then the apparent magnitude of any star located at r > r2 is larger by AV than it would have been without the cloud. Conversely, the radial distances of these background stars, considered as a function of mV, are uniformly lower. According to equation (2.12), these distances increase with mV as
This linear relation holds only in front of the cloud or beyond it. In between, there must be a kink.
There is no practical value to writing the full equation relating log r to mV, since the first quantity is not observed directly. Within any radial interval Ar, however, the number of stars included in Qc is
Thus, if N represents dN*/dmV, the observed number of stars per interval of apparent magnitude, Equations (6.10) and (6.11) imply that N is proportional to r3 for r <r1 or r > r2. Thus, log N = 3 log r to within an additive constant. A plot of the two observable quantities log N and mV should therefore show the same features as the log r — mV relation- a smooth initial rise, a temporary break in slope between two magnitudes m1 and m2 corresponding to r1 and r2, respectively, and then a resumption of the initial slope. Figure 6.3 indicates how AV may be read directly from such a plot.
In practice, one does not have the luxury of observing identical stars with a uniform spatial distribution. However, a generalized method based on the same principal does establish the
Apparent Magnitude mv mj m2
Apparent Magnitude mv
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