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Assume a single-temperature optically thin local thermodynamic equilibrium population, negligible line absorption/stimulated corrections, and that the measured line luminosity is from the remnant's CO. Each spontaneous $J=1\to 0$ decay emits $h{\nu }_{10}$, givingFor a rigid rotor, $E_J/k_B=T_{r}J(J+1)$ with $T_r=h{\nu }_{10}/(2k_B)=2.767$ K. The degeneracy is $g_J=2J+1$, and the rotational partition function isAt 20 K the direct sum gives $Z\simeq 7.57$, $f_1=0.3005$; the approximation $Z\simeq T/T_r+1/3$ is already adequate. Thus, with molecular mass $m_{\mathrm{CO}}\simeq 28m_u$,The remnant age does not enter this optically thin population count. It becomes relevant when estimating emitting volume, density, expansion and optical depth to test the assumptions.

Raising the temperature populates higher rotational states and reduces the fraction in $J=1$. At $T\simeq 101$ K, the rotational partition function gives $f_1\simeq 0.0768$, so the same low-$J$ luminosity requiresabout $3.9$ times the 20-K estimate. Applying the independently computed $J=3$ fraction to the measured high-$J$ line gives the same mass within the model, providing a consistency check rather than an independent proof of its assumptions.

Optical depth, density-dependent excitation, multiple temperature components, beam dilution, aperture mismatch, line blending and flux calibration all need investigation. For one optically thick isothermal source with equal emitting area and velocity width, the Rayleigh-Jeans limit instead gives line-energy luminosity approximately proportional to ${\nu }^3$, so the 3–2/1–0 ratio is of order 27, far below 185. That illustrates why optical-depth and emitting-area assumptions cannot be ignored; different source areas or excitation structures can change this simple comparison. Optically thick clumps can hide mass, and a high-temperature component may dominate the 3–2 line while colder material contributes much of the 1–0 emission.

Additional rotational lines, especially the available 2–1 transition and higher-$J$ lines, test a one-temperature excitation diagram. Less abundant isotopologues help constrain opacity; resolved imaging and line profiles constrain emitting volume and homologous expansion, using the remnant age. Density information permits non-LTE statistical-equilibrium/radiative-transfer modelling. Dust emission can provide a complementary mass constraint with its own opacity and dust-to-gas uncertainties. A total gas mass additionally needs a justified CO abundance, accounting for molecular formation, destruction and unobserved material.

A rigid rotor restricted to one angular coordinate $\varphi$. Single-valued spatial states $e^{im\varphi }$ have $m\in \mathbb{Z}$ and energy $E_m={\hslash }^2{m}^2/(2I)$, with moment of inertia $I$. The zero-angular-momentum level has state degeneracy one; every nonzero $|m|$ has two directions. For labelled molecular ends its high-temperature rotational partition function is $\sqrt{\pi T/{\theta }_{\mathrm{rot}}}$, where ${\theta }_{\mathrm{rot}}={\hslash }^2/(2I{k}_{\mathrm{B}})$. Equivalent orientations and nuclear spin statistics require a further molecular counting convention. One quadratic angular momentum gives rotational heat capacity $k_{\mathrm{B}}/2$ per molecule, rather than $k_{\mathrm{B}}$ for an unrestricted linear rigid rotor.

The rotational partition function sums Boltzmann factors over rotational energy levels, including their degeneracies and the appropriate indistinguishability convention. For a linear rigid rotor at high temperature, $q_{\mathrm{rot}}=8{\pi }^{2}IkT/(\sigma h^2)$, with moment of inertia $I$ and rotational symmetry number $\sigma$.