= Optically thin CO rotational thermometry
{title2=$L_{32}/L_{10}=C\exp[-(E_3-E_1)/(k_BT)]$}
For one optically thin LTE carbon-monoxide component, $L_{ul}=N_uA_{ul}h\nu_{ul}$ and $N_J\propto(2J+1)e^{-E_J/(k_BT)}$. 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.
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