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The Two-Zone Temperature Distribution Model: Inferences on the Structure and Composition of Dusty Protoplanetary Disks

William Grimble, Joel Kastner, B. Sargent, Keivan Stassun

TL;DR

This work introduces a refined Two-Zone Temperature Distribution (TZTD) framework that derives a compact, radially varying thermal-emission formulation for protoplanetary disks and applies it to Spitzer IRS spectra of transition disks. By parameterizing the temperature and surface-density relations with a single distribution parameter $x= rac{2+p}{q}$ and offering a special-case simplification at $x=-1$, the model delivers robust, fast fits to mid-IR data and yields insights into inner-dim cavity sizes and mineralogy. Across a set of disks, the TZTD analysis reveals that cool atmospheric layers often harbor crystalline silicates (notably forsterite) while warm zones exhibit silica-poor, olivine- and pyroxene-rich compositions, implying significant grain processing and growth. The authors discuss the model's strengths and limitations, compare results with previous work, and outline directions to couple TZTD outputs with full radiative-transfer analyses (EaRTH Disk Model) and future JWST/PRIMA observations, enabling a cohesive, multi-wavelength understanding of disk structure and composition.

Abstract

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-IR spectral model that conveyed spatial information and worked in tandem with a radiative transfer model, which formed the EaRTH Disk Model. In this article, we take the empirical portion of that model, the TZTD 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 $Spitzer~Space~Telescope$ Infrared Spectrograph (IRS) 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.

The Two-Zone Temperature Distribution Model: Inferences on the Structure and Composition of Dusty Protoplanetary Disks

TL;DR

This work introduces a refined Two-Zone Temperature Distribution (TZTD) framework that derives a compact, radially varying thermal-emission formulation for protoplanetary disks and applies it to Spitzer IRS spectra of transition disks. By parameterizing the temperature and surface-density relations with a single distribution parameter and offering a special-case simplification at , the model delivers robust, fast fits to mid-IR data and yields insights into inner-dim cavity sizes and mineralogy. Across a set of disks, the TZTD analysis reveals that cool atmospheric layers often harbor crystalline silicates (notably forsterite) while warm zones exhibit silica-poor, olivine- and pyroxene-rich compositions, implying significant grain processing and growth. The authors discuss the model's strengths and limitations, compare results with previous work, and outline directions to couple TZTD outputs with full radiative-transfer analyses (EaRTH Disk Model) and future JWST/PRIMA observations, enabling a cohesive, multi-wavelength understanding of disk structure and composition.

Abstract

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-IR spectral model that conveyed spatial information and worked in tandem with a radiative transfer model, which formed the EaRTH Disk Model. In this article, we take the empirical portion of that model, the TZTD 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 Infrared Spectrograph (IRS) 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.

Paper Structure

This paper contains 25 sections, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Two-Zone Temperature Distribution (TZTD) Model Formulation
  3. Derivation of Thermal Emission
  4. Exact Thermal Emission Assuming x=-1
  5. Application of the TZTD Model to Transition Disk Observations
  6. Data Analysis
  7. Goodness of Fits
  8. Fitted Temperatures and Mineralogies
  9. Comparisons of Mineralogical Results with Previous Work
  10. MP Mus
  11. DM Tau
  12. DoAr 44
  13. GG Tau AA Ab
  14. GM Aur
  15. IP Tau
  16. ...and 10 more sections

Figures (3)

  • Figure 1: Model fit plots of TZTD empirical mineralogical analysis of $Spitzer$ IRS spectra of targets indicated in Table \ref{['targets']}; Red: Cool disk component constituents, Blue: Warm disk component constituents; see Figure Set A1 in Appendix \ref{['appe']} for legend of dust components
  • Figure 2: continued.
  • Figure 3: Results of TZTD empirical mineralogical analysis of MP Mus $Spitzer$ IRS spectrum; (a) Dust composition of optically thin portions of protoplanetary disk, using values from Table \ref{['minres']}, (b) Logarithmic scaling of corresponding model fit plot in Figure \ref{['res_plot']} to demonstrate the contributions of each mineral; (c) Residual between best fit model and $Spitzer$ IRS spectrum; (d) Legend for the model plots, using shortened names from Table \ref{['opacity']}; Red: Cool disk component constituents, Blue: Warm disk component constituents.