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

TL;DR

The paper investigates whether starlight-driven shadows and thermally induced staircases can arise in protoplanetary disks by solving radiation hydrodynamics with frequency-dependent irradiation and flux-limited diffusion. Using 2.5D simulations across varied dust contents and surface densities, the authors find that optically thick, slow-cooling disks can develop long-lived bumps and shadows within ~30 au, forming quasi-steady staircases; however, dust settling can erase these structures, and inner-disk opacity profiles strongly influence the innermost features. The results highlight that dynamic radiative transport and self-consistent dust physics are essential to accurately predict and interpret disk substructures, with implications for observed rings and shadows in scattered light and mm emission. Overall, the work clarifies the conditions under which starlight-driven staircases persist and emphasizes the need for advanced dust-gas radiative modeling to connect theory with observations.

Abstract

Protoplanetary disks observed in millimeter continuum and scattered light show a variety of substructures. Various physical processes in the disk could trigger such features -- one of which that has been previously theorized for passive disks is the thermal wave instability -- the flared disk may become unstable as directly illuminated regions puff up and cast shadows behind them. This would manifest as bright and dark rings, and a staircase-like structure in the disk optical surface. We provide a realistic radiation hydrodynamic model to test the limits of the thermal wave instability in irradiated disks. We carry out global axisymmetric 2D hydrostatic and dynamic simulations including radiation transport with frequency-dependent ray-traced irradiation and flux-limited diffusion (FLD). We found that starlight-driven shadows are most prominent in optically thick, slow cooling disks, shown by our models with high surface densities and dust-to-gas ratios of sub-micron grains of 0.01. We recover that thermal waves form and propagate inwards 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 -- indicating a long-lived staircase structure. We find that existing thermal pressure bumps could produce and enhance this effect, forming secondary 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. We conclude that one can recover thermally induced flared-staircase structures in radiation hydrodynamic simulations of irradiated protoplanetary disks using flux-limited diffusion. We highlight the importance of modeling dust dynamics consistently to explain starlight-driven shadows.

Figures (16)

• Figure 1: Schematic to illustrate the different terms: "bump", "shadow" and "staircase" defined in the introduction. The star is denoted by the orange blob, and the stellar rays hit the irradiation absorption surface $\tau_\star = 1$, at height $z_s = z~(\tau_\star=1)$ from the midplane, and a flaring angle $\varphi$.
• Figure 2: Top panel: Frequency-dependent dust absorption opacities (per gram of dust, $\text{g}_\text{dust}$) from tabulated values 2020AKriegerWolf2022KriegerWolf. The blue points denote 132 frequencies logarithmically sampled in the range $\nu \in [1.5 \times 10^{11}, 6 \times 10^{15}]$ Hz for the computation of irradiation flux in our models. The black line indicates the black-body function and the orange line is the Planck opacity for $T_\star = 4000~$K 2010Kuiper. Bottom panel: Temperature–dependent Planck and Rosseland mean opacities used for radiation transfer. The local maximum ($\approx$ 250 K) and the subsequent dip in the mean opacities is due to the $10\,\mu$m silicate feature.
• Figure 3: Top: Time evolution of the temperature profile at the disk midplane for iterations $i=[10,15,20,25,30]$. The profiles indicate that thermal waves form and travel inwards with time. There are two bumps/shadows at the end of the hydrostatic run for model MFID. The blue dotted line is the radiation temperature at $i=30$ and the black dashed line indicates the background power law fit for the midplane $T_{\text{law}} = 185\,(R/\text{au})^{-0.5}$. Bottom: Deviation of the temperature profile from the background power law.
• Figure 4: 2D profiles of temperature and the irradiation heating term for the hydrostatic model at $i = 30$ iterations. The orange line refers to the $\tau_\star$ = 1 profile for the stellar radiation using the Planck mean opacity. Blue lines refer to $\tau_\nu = 1$ profiles for for different frequencies $\nu$ (same as in Fig. \ref{['fig:opacfig']}), with the line opacity weighted by $B_\nu$ at that frequency. 2022FuksmanKlahr . The red line indicates the $\tau_\text{disk} = 1$ profile for the disk's radiation using $\kappa_P (T)$. The green lines indicate the midplane gas scale height of the disk.
• Figure 5: 2D profiles of temperature and the irradiation heating term for the hydrodynamic model at $t = 500$ orbits (top) and $t = 10000$ orbits (bottom). We see that the inner staircase (corresponding to the hydrostatic model in Fig. \ref{['fig:timeevolstat']}) has grown and slightly shifted inwards with the outer staircase readjusting as a consequence of the first. The black solid line denotes the temperature corresponding to the opacity transition in Fig. \ref{['fig:opacfig']}. We see that these staircases remain in the disk for thousands of orbits spanning several radiative diffusion timescales.