2025ApJ...995...86G

In order to understand the mineralogy and structure of protoplanetary disks, it is important to analyze them from both an empirical spectrum-based perspective and a radiative transfer image-based perspective. In a prior paper, we set forth an empirical mineralogy mid-infrared spectral model that conveyed spatial information and worked in tandem with a radiative transfer model, which formed the Empirical and Radiative Transfer Hybrid (or EaRTH) Disk Model. In this article, we take the empirical portion of that model, the two-zone temperature distribution model, and refine it with a newly derived protoplanetary disk thermal emission formulation which uses a temperature distribution without requiring discrete integration. This simplified model uses an empirical relation between spatial distribution variables, which permits radiative transfer models to directly fit these spatial distribution variables more freely within the provided empirical constraints. We test this model against several Spitzer Space Telescope Infra-Red Spectrograph spectra, primarily transition disks, and discuss the mineralogical and structural implications of the fits, including the implications for grain growth and processing within the atmospheric zones of the disks.