= Solution
\b[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 is
$$
Q_H\simeq\int n_en_p\alpha_B(T_e)\,dV,
$$
whereas 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.
\b[Nebular continua and <recombination lines>.] Their strength depends on the <emission measure>, roughly $\int n_en_i\,dV$, <electron temperature>, ionic abundances and radiative branching probabilities. For a <recombination line>,
$$
L_{ul}\simeq h\nu_{ul}\alpha^{\rm eff}_{ul}(T_e)\int n_en_{\rm ion}\,dV.
$$
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 $j_\nu^{\rm ff}\propto n_en_iT_e^{-1/2}\exp[-h\nu/(k_BT_e)]g_{\rm ff}$; 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>.
\b[Collisionally excited lines.] Below a line's <critical density of an atomic transition>,
$$
j_{ul}\simeq n_en(X^i)q_{lu}(T_e)h\nu_{ul},\qquad
q_{lu}\propto T_e^{-1/2}\exp[-\Delta E/(k_BT_e)].
$$
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.
\b[Dust and molecules.] <Nebular dust spectral features> identify the carriers as well as contributing to the integrated <luminosity>. In an optically thin dust component,
$$
F_\nu\propto\int \kappa_\nu B_\nu(T_d)\,dM_d.
$$
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 $T_d\propto L_*^{1/4}r^{-1/2}$; 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.
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