Because its critical density is relatively low, CO has most often been used for studying massive clouds, rather than the dense cores within them. In these larger regions, the J = 1 ^ 0 transition of12C16O is almost always optically thick, while the same line from rarer isotopes is frequently not. The issue of optical thickness plays a key role when interpreting observations from this molecule.
Figure 6.1 shows three representative J =1 ^ 0 line profiles, all from the same region in Taurus-Auriga. As usual, we plot the antenna temperature TA instead of Iv itself, and the line-of-sight velocity Vr in place of the frequency v. The 12C16O profile in the figure has a flat-topped or saturated appearance, the characteristic sign of optical thickness. Any photon emitted near the line center va is very quickly absorbed by nearby 12C16O molecules. Because the absorbers have a finite relative velocity, the reemitted photons are slightly Doppler-shifted in frequency. As this process is repeated a number of times, photons diffuse in frequency into the line wings, i. e., the original emission profile is broadened.
Most molecules in the cloud have a small relative speed. Thus, 12 C16O photons are still optically thick over some frequency range centered on va. Within this range, the cloud radiates from its surface like a blackbody. The observed intensity Iv is proportional to the Planckian function Bv evaluated at the excitation temperature near the surface (see Appendix C). This function varies little over the relatively narrow frequency range in which the radiation is optically thick. Hence, the profile appears flat. Sufficiently far from line center, there are few enough absorbing molecules that the photons can escape, and the intensity drops.
For more optically-thin transitions, such as the 13C16O and 12C18O lines also shown in Figure 6.1, the profile is reduced in amplitude and more sharply peaked. In these cases, every
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Velocity Vr (km s~1)
Figure 6.1 Intensity profiles of the J = 1 ^ 0 line in three CO isotopes, observed toward Taurus-Auriga.
molecule along the line of sight contributes to the emission, so that the intensity integrated over all frequencies (or, equivalently, TA integrated over Vr) is proportional to the total column density of the isotope in question. It is important to understand that this proportionality cannot hold for 12C16O, whose radiation emanates only from the cloud's surface layers.
To see the matter more quantitatively, consider the ratio of the column densities of the two species 12C16O and 13C16O. Since the oxygen nuclei are identical, this number in nearby clouds should approximately equal the terrestrial value for the ratio of carbon isotopes, which we denote as [12C/13C]*. This ratio is measured to be 89. We do not expect the equality to be exact, because a number of chemical reactions in clouds slightly favor the rarer isotope. Since 13C16O is more tightly bound than 12C16O by 3.0 x 10_3 eV, i. e., by an equivalent temperature of 35 K, such chemical fractionation is significant at molecular cloud temperatures. In warmer clouds, the inferred values of [12C/13C] do fall reasonably close to the terrestrial figure, although there is evidence for a systematic decline toward the Galactic center. For the example shown in Figure 6.1, however, the measured ratio of J* TAdVr for the two lines is only 2.2, much too small to be caused by chemical fractionation alone. The true explanation is that, for the optically-thick 12C16O emission, J* TAdVr does not trace the full column density, which must be inferred by other means.
The general problem we are addressing is how to use the received intensity of any spectral line to deduce the physical conditions within a cloud. Let us focus first on temperature and density as the quantities of interest. A key relation, derived in Appendix C, is the detection equation:
Tb0 = T0 [f (Tex) - f (Tbg)] [1 - exp(-Ar0)] . (6.1)
Here, At0 and TBo are, respectively, the cloud's optical thickness at line center and the received brightness temperature at the same frequency. As discussed in Appendix C, TB (v) is related to the directly observed TA(v) through the beam efficiency and the beam dilution factor. The quantity Tbg in (6.1) is the blackbody temperature associated with any background radiation field, assumed here to be approximately Planckian in its energy distribution. Finally, the function f (T) is defined by f (T) = [exp(To/T) - If1 . (6.2)
Here T0 is the equivalent temperature of the transition, i. e., T0 = hv0/kB.
Let us apply the detection equation to the profiles in Figure 6.1. For this particular observation, both the beam efficiency and dilution factor happen to be close to unity, so that TB « TA. Consider first the 13C16O line. While TB3o is known directly from the observed profile, equation (6.1) still contains, beside Tbg, the two unknowns T^X and At010; here we use the superscript to specify the carbon isotope. Interpretation of the 10C16O profile clearly requires more information. For the optically-thick 12C16O line, however, it is generally true that Ar012 ^ 1, so that the rightmost factor in (6.1) reduces to unity. We then have, to a good approximation,
Assuming the radiation source behind the cloud to be the cosmic microwave background, we set Tbg equal to 2.7 K. We also know that T^"2 is 5.5 K. We may now use equation (6.3) to solve for T^ in terms of the observed quantity Tg^. For our profile, Tg^ = 5.8 K, so we find that TX2 = 9.1 K. The 12C16O line is usually so optically thick that the J = 0 and J =1 level populations can be taken to be in LTE, even if the ambient density is less than ncrit (see § 6.2 below). We therefore have a measure of the interior kinetic temperature:
It is also of interest to evaluate the optical thickness of the cloud to the observed spectral lines. In the case of 10C16O, its collisional and radiative transition rates per molecule, and therefore also its value of ncrit, are very close to those of 12C16O. At a fixed ambient density ntot, one must obtain Tex by considering not only ncrit but also the local radiation intensity (see Appendix B). In the absence of a detailed model, it is difficult to assess the amount of radiative trapping, but it is generally safe to take the lower levels of 10C16O to be in LTE if 12C16O is very optically thick. The relative populations in such levels for the two isotopes are then the same, so that we have
This equality finally allows us to apply the detection equation to 10C16O. With T^ known from the observation to be 4.1 K, and T013 = 5.3 K, we find Ar013 = 1.2. The fact that the 10C16O line is only marginally optically thin is consistent with the profile's appearance in Figure 6.1, where we can see the beginning of saturation broadening.
To estimate the optical thickness of the 12C16O line, we first note that both Ar012 and At013 must be proportional to the total column densities of their respective isotopes:
We have already seen, however, that
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