Optically thin CO rotational thermometry ($L_{32}/L_{10}=C\exp [-(E_3-E_1)/(k_{B}T)]$)

For one optically thin LTE carbon-monoxide component, $L_{ul}=N_u{A}_{ul}h{\nu }_{ul}$ and $N_J\propto (2J+1)e^{-E_J/(k_{B}T)}$. Thus $C=7A_{32}{\nu }_{32}/(3A_{10}{\nu }_{10})$ for the 3–2/1–0 luminosity ratio. The temperature then determines the $J=1$ population fraction and hence the CO mass. Multiple excitation components, optical depth or subthermal excitation invalidate a single-temperature inference.