= Solution
\b[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,
$$
n_l n_e q_{lu}=n_u(A_{ul}+n_eq_{ul}),\qquad
\epsilon_{ul}=n_uA_{ul}h\nu_{ul}.
$$
The critical density is $n_{\rm crit}=A_{ul}/q_{ul}$ for this simple level system; multiple radiative and collisional channels require summed rates. At $n_e\ll n_{\rm crit}$ each <collisional excitation> is likely to produce radiation, so $\epsilon_{ul}\propto n_l n_e q_{lu}h\nu$. 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.
\b[A temperature-sensitive ratio.] For a Maxwell-Boltzmann electron distribution,
$$
q_{lu}(T)\propto T^{-1/2}\Upsilon_{lu}(T)
\exp[-\Delta E_{lu}/(k_BT)],
$$
where $\Upsilon$ 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 ${}^1D_2$ toward the ${}^3P$ ground term, while the $4363$ angstrom line connects ${}^1S_0$ to ${}^1D_2$. 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,
$$
\boxed{R_{\rm OIII}=\frac{I_{4959}+I_{5007}}{I_{4363}}
\simeq7.90\exp\left(\frac{32900\ {\rm K}}{T_e}\right),
\qquad T_e\simeq\frac{32900\ {\rm K}}{\ln(R_{\rm OIII}/7.90)}.}
$$
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.<Resolving the electron temperature discrepancies in H II regions and planetary nebulae: kappa-distributed electrons by David C. Nicholls and others>[Section 4, equation 18 and the following approximation warning; arXiv:1204.3880v2, PDF pages 5–6]{ref}
\Image[/past-exam-of-the-frontiers-of-observational-astrophysics-course-of-the-university-of-cambridge/2015/paper-1-oxygen-thermometer.png]
{title=Original oxygen forbidden-line level schematic and low-density Maxwellian temperature-ratio curve; level spacings are approximate}
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.
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