a little bit more about fractional molecular abundances

2.4 Practical Issues for Deriving Molecular Abundances
2.4.1 Source Structure
The above formulation contains various assumptions. First of all, the cloud is assumed to be homogeneous in physical and chemical conditions along a line of sight. In reality, physical conditions and molecular abundances vary within a cloud. For instance, H2 density will be lower at the periphery of the cloud. Furthermore, molecular abundances will also vary within a cloud because of different chemical processes in different cloud regions. For example, some molecules are abundant in dense regions; other molecules are abundant in the periphery. Observations trace the emission integrated over these effects along a line of sight.

This situation is further complicated by the limited spatial resolution of tele- scopes. If the source size is smaller than the telescope resolution (beam size), the observed emission is significantly diluted. For instance, we consider a circular source with a diameter of θs with a uniform surface brightness on the celestial sphere. If this source is observed with a Gaussian beam with a full-width at half maximum (FWHM) beam size of θb, the observed intensity becomes weaker by a factor of:
!
1 exp 1:44θ2s : ð2:52Þ θ2b
If the surface brightness distribution is represented by a Gaussian function with an FWHM width of θs, the intensity becomes weaker by a factor of:
θ2s : ð2:53Þ θ 2s þ θ 2b
The decrease in the intensity is called the beam dilution effect. The limited resolution causes an additional problem when we employ the intensity data observed with different beam sizes. The beam size of a telescope depends on the observed frequency (Chap. 11). Hence, we have to be careful when the intensities of two or more lines observed at different frequencies are used to derive the column densities and excitation temperatures. If the source size is comparable to or smaller than the beam size, the correction for the beam dilution effect is very important so as not to introduce systematic error into the interpretation of the data.
Moreover, the source structure can be different for different lines of sight. If the cloud has an internal structure whose size is smaller than the telescope resolution (beam size), the observing emission is “beam averaged.” If the molecular abun- dances vary among small-scale structures, the derived column density is also “beam averaged.”
Fractional abundances are typically derived relative to the H2 molecule for comparison with chemical models. In this case, the observed column density has to be divided by the H2 column density observed with the same beam size. Because we cannot directly observe the H2 molecule in interstellar clouds as it lacks electric dipole-allowed transitions in the radio and infrared regimes, the H2 column density is derived from observations of rare isotopic species lines of CO such as C18O by assuming a relative abundance. However, it is known that CO is depleted onto dust grains in cold and dense parts of clouds, as described in Chap. 6. Hence, the H2 column density derived from the rare isotopic species lines of CO would be underestimated. Furthermore, the critical densities of the CO lines are lower than those of other molecular lines, and hence CO emission tends to trace longer distances along the line of sight than the other molecular lines.

Instead of CO, the millimeter-wave and submillimeter-wave continuum emis- sion of dust grains is often used to derive the H2 column density. In this case, we need to assume the emissivity of dust grains and the gas-to-dust ratio (about 100). In any case, we have to consider that the H2 column density and temperature are again beam averaged. Hence, the derived fractional abundances are also beam-averaged values. If the source is expected to have internal structure unresolved with the telescope beam, we have to be careful in interpreting the abundances.