Density-depth degeneracy of free-free radiation ($n_e\mapsto q{n}_e,L\mapsto L/q^2$)

For a uniform ionized hydrogen column with known temperature and fixed projected angular area, free-free brightness and optical depth depend on $\mathrm{EM}=n_e^{2}L$. 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 $n_{e}L$, can break the degeneracy; gas clumping factor and temperature structure complicate that inference.