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Stellar radiation and nebular ionization. The planetary-nebula spectral energy distribution is constrained by a shared radiation and thermal balance. The central star's effective temperature, luminosity and evolutionary state set its continuum and ionizing-photon supply. A hotter star shifts the ionizing spectrum and allows higher ionization stages; the total recombination output also depends on how many ionizing photons the gas actually absorbs. In an ionization-bounded nebula, approximate photoionization balance iswhereas a density-bounded nebula can leak photons. Dust absorption of ionizing photons changes this relation further. Stellar-atmosphere and wind effects make a blackbody an approximation, especially in the extreme ultraviolet. A changing core temperature and expanding envelope explain why nebular spectra evolve.

Nebular continua and recombination lines. Their strength depends on the emission measure, roughly $\int n_e{n}_{i}dV$, electron temperature, ionic abundances and radiative branching probabilities. For a recombination line,Case-B line ratios assume appropriate trapping of Lyman-series photons and escape of the lower-series radiation. Density, optical-depth effects and extinction can alter observed ratios. The free-free emissivity has the schematic dependence $j_{\nu }^{\mathrm{ff}}\propto n_e{n}_i{T}_e^{-1/2}\exp [-h\nu /(k_B{T}_e)]g_{\mathrm{ff}}$; the nebular free-bound continuum also carries ionization thresholds and recombination cross-sections. Two-photon emission depends on the population of the hydrogen $2s$ state and can be reduced by collisional mixing into $2p$. Density clumping matters because these rates are weighted approximately by density squared, not simply by total gas mass.

Collisionally excited lines. Below a line's critical density of an atomic transition,Elemental abundance, ionic fraction and electron temperature therefore all enter. The ionizing spectrum and local ionization parameter help determine which ion is present. Above the critical density of an atomic transition, collisional de-excitation competes with radiative decay and can suppress a forbidden line. Low-energy infrared fine-structure lines are generally less exponentially temperature sensitive than high-excitation optical/ultraviolet lines. Metal-line cooling also helps set the temperature, so changing an abundance changes both the number of emitters and the thermal balance; line luminosity need not be proportional to abundance at fixed everything else.

Dust and molecules. Nebular dust spectral features identify the carriers as well as contributing to the integrated luminosity. In an optically thin dust component,The grain mass sets its normalization, while the radiation field, stellar distance, grain size and absorption efficiency determine its temperature distribution. For approximately grey heating, radiative equilibrium suggests $T_d\propto L_{\ast }^{1/4}{r}^{-1/2}$; non-grey grains change this scaling. Long-wavelength opacity is often approximated as ${\kappa }_{\nu }\propto {\nu }^{{\beta }_d}$. Small aromatic grains undergo transient heating, so one equilibrium temperature is not adequate for every infrared feature. Carbon-to-oxygen chemistry, grain processing and shielding affect aromatic versus silicate bands. Molecular line emission further depends on molecule abundance, collisional density, kinetic temperature, ultraviolet pumping and shocks, while shielding and dissociation set whether molecules survive at all.

Finally, interstellar extinction preferentially suppresses ultraviolet and optical light, and dust reradiates some absorbed energy in the infrared. Aperture, angular resolution and distance must be treated consistently before comparing stellar, nebular and dust components. These dependencies make the combined spectrum an energy and abundance diagnostic, but also prevent a unique reconstruction from a single broad-band colour.

A spontaneous emission transition has probability per unit time $A_{ji}$ and emits a photon of energy $h{\nu }_{ji}$. For an optically thin gas, its angle-integrated emissivity is $j_{ji}=n_j{A}_{ji}h{\nu }_{ji}$. Electron collisions populate excited levels at rates $n_e{n}_i{q}_{ij}$; they also remove excitation collisionally at rates $n_e{n}_j{q}_{ji}$, returning the energy to the gas rather than producing the line photon. A collisionally excited line is usually emitted by a subsequent spontaneous radiative decay; collisional de-excitation instead quenches it.

The critical density of an atomic transition compares radiative loss from an upper level with its collisional loss. For a two-level model, $n_{\mathrm{crit}}=A_{ji}/q_{ji}$; for several lower or connected levels use the relevant sums. Below this density, excitation is commonly followed by a photon and the line cools the gas. Above it, collisions increasingly suppress radiative emission per excitation and drive the level populations toward Boltzmann ratios.

Optical forbidden emission lines have small $A$ because an electric-dipole transition is forbidden by a selection rule. Metastable levels survive long enough to radiate in dilute ionized nebulae, although collisions quench them in dense gas. Ratios from closely spaced levels with different critical densities are electron-density diagnostics; ratios involving different excitation energies probe the electron temperature through Boltzmann factors. Combining line emissivities with electron density, temperature and an ionization model estimates ionic abundances.

Infrared fine-structure emission lines connect closely spaced sublevels. Their small excitation energies make them useful over temperature ranges where optical excitation is exponentially suppressed, and dust extinction is usually weaker in the far infrared. They are important coolants and density diagnostics in ionized gas, with the choice of ion specifying the ionization zone. Neither an optical nor an infrared ionic abundance automatically equals the total elemental abundance; unobserved ionization stages require an ionization correction.

The infrared and optical transitions share the same upper level 3. Using the supplied infrared photon energy givesWith the same $T_e,n_e$ and three-level fraction $f_3$ used in the abundance calculation of b,This exceeds the optical abundance by a factor $5.53$. Because the upper level is common, its population cancels in the intrinsic emissivity ratio:The observed ratio is $1.5\times 10^{-5}$. Thus changing only the electron temperature or a common level-population correction cannot reconcile these two lines from the same gas. Dust extinction suppressing the optical line is a natural explanation. If the far-infrared attenuation is negligible, the required optical attenuation isDifferent apertures, filling factors, optical-depth assumptions or emissivity calibrations could also cause a discrepancy. A total sulfur abundance would still need an ionization correction beyond this ionic comparison.