= Solution
Raising the <temperature> populates higher rotational states and reduces the fraction in $J=1$. At $T\simeq101$ K, the <rotational partition function> gives $f_1\simeq0.0768$, so the same low-$J$ <luminosity> requires
$$
\boxed{M_{\rm CO}\simeq7.36\times10^{-2}M_\odot},
$$
about $3.9$ 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.
\Image[/past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015-co-rotational-mass-temperature.png]
{title=Temperature-dependent CO line ratio and mass inferred from the fixed J=1 to 0 luminosity under optically thin LTE}
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