Planetary nebula

The spectral energy distribution of a planetary nebula combines hot central-star radiation, nebular free-free/free-bound/two-photon continua, recombination and collisionally excited lines, infrared dust emission and sometimes molecular emission. Ionizing-photon supply, gas thermal/ionization balance, density, composition, geometry and extinction determine their relative strengths. A line-dominated broad-band flux is not by itself a continuum temperature measurement.

Dust contributes a broad infrared continuum together with composition-dependent vibrational structure. Carbon-rich material may show aromatic bands, while oxygen-rich silicates can show features near 9.7 and 18 micrometres. Grain size, charging, transient heating, optical depth and chemical processing affect these features, so one dust temperature cannot characterize every band carrier.

A nebular free-bound continuum is emitted when a free electron is captured into a bound atomic level and its excess energy leaves as a photon. The threshold energies produce recombination edges such as the Balmer discontinuity. Its spectrum depends on electron temperature, ionic abundances, capture cross-sections and emission measure; it is distinct from bound-bound recombination lines and free-free bremsstrahlung.

The Zanstra method estimates a central star's effective temperature by comparing its observed nonionizing continuum with the absorbed ionizing-photon rate inferred from nebular recombination lines. If ${\Phi }_{E_0}(T)={\int }_{E_0/h}^{\infty }{B}_{\nu }(T)/(h\nu )d\nu$, then $[F_{\ell }/(h{\nu }_{\ell }{F}_{{\nu }_o,\ast })][\beta /{\alpha }_{\ell }^{\mathrm{eff}}]={\Phi }_{E_0}(T_s)/B_{{\nu }_o}(T_s)$ when the line traces the full appropriate ionizing budget. Both the distance and stellar radius cancel. Density inhomogeneity also cancels from the integrated recombination estimate when the coefficient ratio is known; photon leakage, dust and mixed-species transfer can invalidate the simple budget.