We describe a non-LTE photoionization code that calculates the atmospheric structure and emergent spectrum of a red giant illuminated by the hot component of a symbiotic binary system. The model assumes hydrostatic, radiative, and statistical equilibrium for the red giant atmosphere and solves the radiative transfer equation with a local escape probability method. We compute non-LTE level populations for a variety of ions and predict the variation of emission-line fluxes as function of the temperature and luminosity of the hot component. Our models produce strong emission lines only when the hot component has a high effective temperature, Th 100,000 K, for hot component luminosities, Lh 630 L☉. Predicted electron densities and temperatures for the photoionized atmosphere agree with observations. The models also produce reasonably large continuum variations that are consistent with the light curves of some symbiotic stars. However, predictions for most optical and ultraviolet emission-line fluxes fall well below those observed in typical symbiotic stars. We conclude that the hot component must illuminate a red giant wind to reproduce observed line fluxes. Hydrostatic red giant atmospheres simply do not have enough material beyond the photosphere to account for the emission features observed in most symbiotics. Illumination can modify the structure of a red giant atmosphere even when the emitted spectrum changes very little. Energetic photons from the hot component cause the atmosphere to expand by several percent for large hot component luminosities. This expansion is insufficient to increase the red giant mass-loss rate, except in systems where the giant already fills or nearly fills its Roche lobe.
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