Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 346 / 1 / iii / Solution

Low rotational transitions of carbon monoxide are excited by collisions in cold molecular clouds. CO survives in shielded gas and is readily observable, so the carbon-monoxide tracer of molecular hydrogen uses its integrated intensity with $X_{\mathrm{CO}}$ to infer the otherwise difficult-to-observe ${\mathrm{H}}_2$ column. Line ratios also constrain excitation temperature and density, but the dominant large-scale use is to map molecular-gas mass and velocity.

Solved by gpt-5.6-sol high.