X-factor

Since the H$_2$ molecule is the homonuclear diatomic molecule, the H$_2$ molecule has no electric dipole. Thus, the electric dipole radiation is not expected for this molecule. While the second abundant molecule CO is the heteronuclear diatomic molecules and has electric dipole. Lower levels of rotational transition of CO are relatively easily excited even in the cold interstellar medium of $T\sim 10{\rm K}$. Therefore, the rotational transition of CO molecule is the first possibility to observe cold molecular gas. The $X$-factor represents the ratio of the column density of H$_2$ molecules $N({\rm H}_2)$ to the integrated intensity of $^{12}$CO line $I_{\rm CO}$ as

$$X\equiv\frac{N_{{\rm H}_2}(\rm cm^{-2})}{I_{\rm CO}({\rm K km s^{-1}})}.$$ (F.1)

There are several empirical but physical estimations of the $X$-factor: Using $\gamma$-rays, $X$-factor is estimated as $X \simeq 2.3 \times 10^{20}$(Strong et al. 1988). This is based on the idea that the emissivity of $\gamma$-ray emission is proportional to the cosmic-ray intensity times the number of target nuclei. The $\gamma$-ray intensity is proportional to the column density if the cosmic-ray intensity is uniform. From the distribution of $\gamma$-ray intensity, Strong et al. (1988). estimated total column density $N{\rm H}_2(\rm cm^{-2})$ using some model. This gives the value of $X$-factor.

Estimations using the Virial mass of the interstellar cloud and the visual extinction $A_V$.