= Solution
<Forbidden emission lines> arise from transitions strongly suppressed in the electric-dipole approximation, often between metastable levels. They can radiate in dilute gas because collisions are infrequent enough for a slow radiative decay to occur. In a two-level optically thin approximation, statistical equilibrium gives
$$
n_un_e q_{ul}+n_uA_{ul}=n_ln_eq_{lu},\qquad
j_{ul}=n_uA_{ul}h\nu.
$$
The <critical density of an atomic transition> is $n_{\rm crit}=A_{ul}/q_{ul}$. At densities well below it, almost every collisional excitation radiates; above it, collisional de-excitation quenches the line relative to its low-density scaling. Collision coefficients and level populations determine the detailed multi-level diagnostic.
To measure <temperature>, compare lines of the same ion with significantly different upper-level excitation energies. The excitation coefficient contains a factor roughly $T^{-1/2}\exp[-\Delta E/(k_BT)]$ times a collision strength. The <oxygen forbidden-line temperature diagnostic> compares the relatively high-excitation $[\mathrm{O\,III}]$ $4363$-angstrom line with the strong $4959$- and $5007$-angstrom lines. Their ratio is sensitive to <electron> <temperature> while canceling much of the common ionic abundance and emission measure.
To measure density, use upper levels with similar excitation energies but different radiative lifetimes and critical densities, as in $[\mathrm{S\,II}]$ $6716/6731$. Their relative collisional quenching changes the ratio over an intermediate density range. At very low and very high densities the ratio approaches limiting values and loses sensitivity. A practical analysis fits density and <temperature> together, corrects wavelength-dependent extinction and calibration, checks blends and optical-depth assumptions, and uses reliable atomic transition and collision data. A single measured ratio generally constrains an emitting-gas model rather than proving every point of a nonuniform nebula has the same <temperature>.
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