Emission line ($I_{\mathrm{line}}>I_{\mathrm{continuum}}$)

An emission line is a localized excess of specific intensity above the continuum. Heated chromospheric material can give a line source above the photospheric continuum source. Optically thin circumstellar gas can also add line photons through recombination or collisional excitation. Such line formation need not obey a thermal photospheric source prescription.

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.

In a three-level ion, statistical equilibrium must include collisional coupling between both upper levels as well as excitations from the ground. The upper populations solve a two-by-two linear system when expressed relative to the ground population. Their line ratio is a rational function of electron density, interpolating between a collisionally excited low-density limit and a collisional Boltzmann limit. Closely spaced upper energies reduce temperature sensitivity, but neglecting upper-level collisional mixing can bias the inferred density.

For low-density gas with Maxwellian electrons, the excitation-energy difference between the ${}^1{S}_0$ and ${}^1{D}_2$ levels of doubly ionized oxygen makes $(I_{4959}+I_{5007})/I_{4363}$ temperature sensitive. A constant-collision-strength approximation is $R\simeq 7.90e^{32900\mathrm{K}/T_e}$. Increasing temperature strengthens the higher-excitation auroral line and lowers $R$. Density, extinction and nonthermal electrons require separate treatment; the $5007/4959$ pair shares one upper level and is primarily a branching ratio rather than this thermometer.