= Density-depth degeneracy of free-free radiation
{title2=$n_e\mapsto qn_e,\quad L\mapsto L/q^2$}
For a uniform ionized hydrogen column with known <temperature> and fixed projected angular area, free-free brightness and <optical depth> depend on $\mathrm{EM}=n_e^2L$. The displayed rescaling leaves the whole ideal spectrum unchanged. <density> and path length cannot be separated without additional information. A known geometry/distance, or a distinct column diagnostic proportional to $n_eL$, can break the degeneracy; <gas clumping factor> and <temperature> structure complicate that inference.
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