 Solution

For optically thin CO rotational thermometry of the same emitting population, the upper-level abundance ratio isThe difference $E_3-E_1$ can be obtained directly from the two successive transition frequencies, $h({\nu }_{32}+{\nu }_{21})$, or approximately from $10k_B{T}_r$. Using the supplied rounded molecular data gives $(E_3-E_1)/k_B=27.66$ K. ThereforeThe observed ratio is $2.46\times 10^{26}/(1.33\times 10^{24})=184.96$, henceAt 20 K the predicted ratio would be only about 61, so the two detections reject that single-temperature optically thin 20-K model. The inferred value is an excitation temperature under the stated assumptions, not a model-independent measurement of the gas's kinetic temperature.