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A forbidden emission line is suppressed by the electric-dipole selection rules but can occur through weaker radiative channels such as magnetic-dipole or electric-quadrupole emission. Its relatively long-lived upper atomic level is vulnerable to collisional de-excitation in dense gas. Low-density astrophysical plasmas can therefore display strong forbidden-line emission even when laboratory dense sources do not.

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 givesThe critical density of an atomic transition is $n_{\mathrm{crit}}=A_{ul}/q_{ul}$. 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 $T^{-1/2}\exp [-\Delta E/(k_{B}T)]$ times a collision strength. The oxygen forbidden-line temperature diagnostic compares the relatively high-excitation $[\mathrm{O}\mathrm{III}]$ $4363$-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 $[\mathrm{S}\mathrm{II}]$ $6716/6731$. 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.

Weak transitions in dilute gas. A forbidden emission line is a transition absent in the electric-dipole approximation because of its selection rules, but permitted through weaker channels such as magnetic-dipole or electric-quadrupole radiation, sometimes aided by state mixing. It is not an impossible transition. Its upper atomic level is relatively long lived, so collisions can deactivate it before a photon is emitted in dense gas. Low-density astrophysical plasmas allow radiative decay to compete successfully.

In a schematic two-level statistical equilibrium,The critical density is $n_{\mathrm{crit}}=A_{ul}/q_{ul}$ for this simple level system; multiple radiative and collisional channels require summed rates. At $n_e\ll n_{\mathrm{crit}}$ each collisional excitation is likely to produce radiation, so ${ϵ}_{ul}\propto n_l{n}_e{q}_{lu}h\nu$. At high density collisional de-excitation suppresses the line relative to this low-density expectation. The rate need not decrease absolutely with increasing density; the suppression is relative to the naive excitation scaling.

A temperature-sensitive ratio. For a Maxwell-Boltzmann electron distribution,where ${\rm Y}$ is an effective collision strength. Lines from upper levels with different excitation energies therefore have a strongly temperature-dependent ratio. Using one ion reduces sensitivity to abundance and ionization fraction compared with comparing unrelated species.

A standard example is doubly ionized oxygen: the $4959$ and $5007$ angstrom lines descend from ${}^1{D}_2$ toward the ${}^{3}P$ ground term, while the $4363$ angstrom line connects ${}^1{S}_0$ to ${}^1{D}_2$. The higher excitation threshold makes the auroral $4363$ line relatively stronger in hotter gas. In the low-density Maxwellian approximation, with slowly varying collision strengths,The numerical form is a useful approximation, not a universal exact atomic law; a multilevel atomic calculation with current collision strengths is needed for precise work.^[ref]

In an active galactic nucleus these lines commonly diagnose the relatively dilute narrow-line gas; dense broad-line gas can collisionally quench them. Measure and deblend the fluxes, subtract continuum and stellar features, correct differential extinction, and check density independently. The $5007/4959$ ratio mainly tests radiative branching and calibration because both lines share one upper level; it is not the required temperature thermometer. Temperature gradients, multiple density components, shocks, ionization structure and departures from the Maxwellian electron distribution can bias a single inferred temperature. Without a measured ratio and density regime, no particular AGN temperature is numerically determined here.