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

$$\left(A_{21}+C_{21} n \right)N_2=C_{12} n N_1.$$ (2.155)

Using

$$C_{12}g_1=C_{21}g_2\exp{\left[-(E_2-E_1)/kT\right]},$$ (2.156)

the number fraction of upper level is written

$$\frac{N_2}{N_1+N_2}=\frac{\displaystyle \frac{\displaystyle ... ...tyle g_2}{\displaystyle g_1}\exp{\left[-(E_2-E_1)/kT\right]}},$$ (2.157)

where

$$n_{\rm cr}\equiv \frac{A_{21}}{C_{21}},$$ (2.158)

is called critical density. When $n\gg n_{\rm cr}$, the second term of the denominator is small and the level population is given by the Boltzmann distribution. As long as $n\ll n_{\rm cr}$, 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 $B$-coefficients, which has a meaning of the cross-section for the radiation, is proportional to the electric dipole moment of the molecule, $A$-coefficients are large for molecules with large electric dipole moment (eq.[2.154]). In the case of rotational levels, $A$-coefficients increase $\propto (J+1)^3$ 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 $A$ and $C$ coefficients. Comparing $J=1-0$ transitions of CO, CS, and HCO$^+$, CS and HCO$^+$ trace higher-density gas than CO. And higher transition $J=5-4$ lines trace higher-density gas than lower transition $J=1-0$ 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, $\beta_\nu$, the critical density is reduced to

$$n_{\rm cr}\sim \frac{A_{21}}{C_{21}}\beta_\nu.$$ (2.159)

The most abundant CO, $^{12}$C$^{16}$O, is sometimes optically thick, while rare molecules $^{13}$C$^{16}$O and $^{12}$C$^{18}$O, which have almost the same Einstein's coefficients as $^{12}$C$^{16}$O, are optically thin. In such a cloud, the critical density of $^{12}$C$^{16}$O is smaller than that of $^{13}$C$^{16}$O and $^{12}$C$^{18}$O.

Table 2.1: $A$ and $C$ coefficients and critical densities for rotational transition of typical molecules

	CO	CS	HCO$^+$
$A_{10}({\rm s}^{-1})$	$7.2\times 10^{-8}$	$1.8\times 10^{-6}$	$4.3\times 10^{-5}$
$C_{10} ({\rm cm^{-3} s})$	$2.8\times 10^{-11}$	$3.7\times 10^{-11}$	$2.6\times 10^{-10}$
$n_{\rm cr 10}({\rm cm^{-3}})$	$2.7\times 10^3$	$5\times 10^{4}$	$1.6\times 10^{5}$
$A_{54}({\rm s}^{-1})$	$1.3\times 10^{-5}$	$3.0\times 10^{-4}$	$7.2\times 10^{-3}$
$C_{54} ({\rm cm^{-3} s})$	$9.3\times 10^{-11}$	$3.3\times 10^{-11}$	$4.2\times 10^{-10}$
$n_{\rm cr 54}({\rm cm^{-3}})$	$1.4\times 10^5$	$9\times 10^{6}$	$1.7\times 10^{7}$