The optically thick LTE case

It has been shown earlier (equation 3.6) that the radiation intensity at a frequency $\nu$ observed by a telescope is given by

$$I_{\rm obs}=\left[ I_{T_{\rm EX}} - I_{T_{B}} \right] \left( 1-e^{-\tau}\right)$$ (5.1)

where $T_{EX}$ is the excitation temperature in the cloud (which is assumed to be constant) and $T_B$ is the cosmic background radiation temperature (note that $I_{T_{\rm EX}}=S_\nu$). For a black body

$$I_{\rm T}= \frac{2h\nu^3} {c^2\left( e^{\frac{h\nu}{kT}}-1 \right) }$$ (5.2)

In the Rayleigh-Jeans limit ( $\frac{h\nu}{kT} \ll 1$) $I_{\rm obs}$ can be written

$$I_{\rm obs}=\frac{2k\nu^2 T_A}{c^2}$$ (5.3)

Note that this is the definition of ${\rm T_A}$, in other words the antenna temperature is defined as being the temperature that would be caused by a blackbody radiating in the Rayleigh-Jeans limit. The radiation from the cloud and cosmic background, however, are not necessarily in the Rayleigh Jeans limit and thus are described by equation 5.2. Substituting these into equation 5.1 and cancelling where appropriate yields

$$T_A=\frac{h\nu}{k}\left[\frac{1}{e^{\frac{h\nu}{kT_{EX}}}-1}-\frac{1}{e^{\frac{h\nu}{kT_{B}}}-1}\right]\times(1-e^{-\tau})$$ (5.4)

If the collision rate between molecules in the cloud is very high then the excitation temperature will be approximately equal to the kinetic temperature. Thus by setting the excitation temperature in the cloud to a given value (say 20 K), insuring the collision rate is very high (this is done by setting the hydrogen density to a very high level) and choosing the molecular abundance such that $\tau \gg 1$ in the line centre the precise value of the line peak can be calculated. For $^{13}$CO the temperatures should be

Transition	Temperature (K)
3 $\rightarrow$ 2	13.058
2 $\rightarrow$ 1	14.952
1 $\rightarrow$ 0	16.582

Line profiles with $T_{\rm EX}=20$K, $\tau \gg 1$ in the line centre for J $=3\rightarrow 2$ (left), J $=2\rightarrow 1$ (middle) and J $=1\rightarrow 0$ (right).

The figures above demonstrate that the program successfully reproduces the lines at the temperatures predicted. The flat topped line shape is also as might be expected since the height of the top represents the temperature that will be caused by the emission from molecules at a particular velocity that have their excitation temperature equal to the kinetic temperature of the gas. The width of the line is determined by the random motion of the molecules in the gas. The velocity distribution of this motion is Gaussian and therefore further away from the line centre there will be fewer molecules at a particular velocity. As long as there are sufficient molecules so that $\tau \gg 1$ the emission temperature will still be at the value given by equation 5.4, this causes the flat top. However, once the columnn density drops below the required level the emission temperature drops off. In the above figures the density has been set so high that the optical depth only becomes less than one at the very edges of the lines.