## Critical Density

Consider a two-level atom (a hypothetical atom which has only two levels), in which the spontaneous downward transitions and collisional excitations and deexcitations are in balance as

 (2.151)

Using
 (2.152)

the number fraction of upper level is written
 (2.153)

where
 (2.154)

is called critical density. When , the second term of the denominator is small and the level population is given by the Boltzmann distribution. As long as , the number of upper-level population is much smaller than that expected for the Boltzmann distribution. Such a low-density gas emits only weakly.

Since -coefficients, which has a meaning of the cross-section for the radiation, is proportional to the electric dipole moment of the molecule, -coefficients are large for molecules with large electric dipole moment (eq.[2.150]). In the case of rotational levels, -coefficients increase and thus the critical density increases for higher transition. In Table 2.1, the critical densities for rotational transitions of typical molecules are shown as well as and coefficients. Comparing transitions of CO, CS, and HCO, CS and HCO trace higher-density gas than CO. And higher transition lines trace higher-density gas than lower transition lines.

In the discussion above, we ignored the effect of transition induced by absorption. The above critical density is defined for optical thin case. For a gas element with a finite optical depth, photons are effectively trapped in the gas element (photon trapping). If we use the probability for a photon to escape from the gas element, , the critical density is reduced to

 (2.155)

The most abundant CO, CO, is sometimes optically thick, while rare molecules CO and CO, which have almost the same Einstein's coefficients as CO, are optically thin. In such a cloud, the critical density of CO is smaller than that of CO and CO.

 CO CS HCO

Kohji Tomisaka 2009-12-10