Starlight-driven flared-staircase geometry in radiation hydrodynamic models of protoplanetary disks

¹ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
² Fakultät für Physik und Astronomie, Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
³ Ludwig-Maximilians-Universität München, Universitäts-Sternwarte, Scheinerstraße 1, 81679 München, Germany
⁴ Exzellenzcluster ORIGINS, Boltzmannstr. 2, 85748 Garching, Germany

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 21 August 2025
Accepted: 2 January 2026

Abstract

Context. Protoplanetary disks observed in the millimeter continuum and scattered light show a variety of substructures. While embedded planets are a common explanation, various physical processes in the disk during the early planet formation phase might also trigger these features. One such possibility that has been theorized previously for passive disks is the thermal wave instability or the stellar irradiation instability, where that the flared disk might become unstable as directly illuminated regions puff up and cast shadows behind them. This would manifest as bright and dark rings and as a staircase-like structure in the disk optical surface.

Aims. We provide a realistic radiation hydrodynamic model to test the limits of the thermal wave instability in starlight-heated protoplanetary disks. We make quantitative comparisons to existing results in the literature from simpler linear theory and 1D models to moment-transfer methods to elucidate the importance of correct numerical treatment for this problem.

Methods. We carried out global axisymmetric 2D hydrostatic and dynamic simulations that include radiation transport with frequency-dependent ray-traced irradiation and flux-limited diffusion. We varied the dust-to-gas ratios and surface densities. We also highlight the role of small grains and dust settling with the first radiation hydrostatic dust models to study starlight-driven shadowing.

Results. We found that starlight-driven shadows are most prominent in optically thick, slowly cooling disks, which are shown by our models with high surface densities and dust-to-gas ratios of submicron grains ϵ_dg = 0.01. We found that thermal waves form and propagate inward in the hydrostatic limit. In contrast, our hydrodynamic models show bumps and shadows within 30 au that converge to a quasi-steady state on several radiative diffusion timescales, which indicates a long-lived staircase structure. Existing thermal pressure bumps might produce and enhance this effect and form secondary structures due to starlight-driven shadowing downstream. Hydrostatic models with self-consistent dust settling instead show a superheated dust irradiation absorption surface with a radially smooth temperature profile without staircases.

Conclusions. We conclude that we can recover thermally induced flared-staircase structures in radiation hydrodynamic simulations of irradiated protoplanetary disks using the flux-limited diffusion method. The shadowing effect is sensitive to the dust content in the disk. We highlight the importance of modeling dust dynamics consistently to explain starlight-driven shadows.