Shadow-Induced Warps in Protoplanetary disks

TL;DR

This work shows that shadows cast by a misaligned inner disk can thermally drive global warps in outer protoplanetary disks, with a 30° inclination producing stronger accretion and a pronounced outer warp (tilt up to ~$32^{\circ}$ relative to the inner disk) than polar configurations. Using 3D radiation–hydrodynamical simulations and pure hydro runs with prescribed temperature structures, the authors demonstrate that the warp is driven by an $m=1,n=1$ temperature perturbation, whose influence peaks near a mutual inclination of ~$15^{\circ}$ but remains significant from $3^{\circ}$ to $30^{\circ}$. They establish a semi-quantitative scaling between the $m=1,n=1$ mode amplitude and disk tilt, show that an outer exponential cutoff enhances inter-disk twisting, and reveal periodic tilt oscillations with periods of order $10^2$–$10^3$ years in full-disk models. The findings offer concrete, testable predictions for ALMA and NIR observations, enabling forward modeling of shadow-induced dynamics by directly constraining the azimuthal–vertical temperature structure and comparing it to the disk’s density and velocity responses.

Abstract

Shadows are commonly observed in protoplanetary disks in near-infrared and (sub)millimeter images, often cast by misaligned inner disks or other obscuring material. While recent studies show that shadows can alter disk dynamics, only the case symmetric across the midplane (e.g., from a polar-aligned inner disk) has been studied. Here we study shadows cast by an inner disk with a $30^\circ$ mutual inclination using 3D radiation-hydrodynamical simulations. Given the same shadow shape and amplitude, the $30^\circ$ inclined shadow leads to a much stronger accretion compared with the polar case, reaching $α\sim$ 1, because the disk is squeezed twice in one azimuth, leading to shocks and strong radial flows near the midplane. The outer disk develops a warp: the inner disk region tilts toward alignment with the shadow, while the outer, exponentially tapered disk tilts and twists in a different direction, inclined $\sim$ 32$^\circ$ relative to the inner region. Locally isothermal simulations with a prescribed temperature structure reproduce the effect, confirming that it is thermally driven. Fourier-Hermite analysis shows that it is the m=1, n=1 temperature perturbation that drives the warp by launching bending waves, with the tilting response of the disk approximately proportional to the modal amplitude. This mode always exists unless the shadow is coplanar or polar. Given a fixed temperature contrast, the m=1,n=1 mode peaks at $\sim$15$^\circ$ mutual inclination, but still contributes substantially across 3$^\circ$ to 30$^\circ$. Shadows cause disk warps--they are not only a consequence of them. We discuss testable predictions for current and future ALMA and NIR observations.

Figures (20)

• Figure 1: Time evolution of the 3D hydrodynamical model (R30), showing the density on the midplane surface at $t=0$, $100$, and $500\ P_0$ (left to right, $P_0 \approx$ 253 yr). The stellar irradiation field is prescribed as a spherical distribution with a shadow lane inclined by $30^\circ$ relative to the original disk midplane (sketch in the top left corner, Equation \ref{['eq:irradiation']}). The disk develops a warp in response to the shadow and also launches two-armed spirals (visible in the zoomed-in panel). The inner and outer disks warp in different directions, traced by the dotted lines marking the twist angle $\gamma$ along different radii. Concentric grid lines are spaced every 160 au starting from $R=160$ au, and azimuthal divisions are spaced every $22.5^\circ$. The dashed gray lines are $\phi=0, \pi/2, \pi$, and $3\pi/2$ in the original coordinate. Associated animation for this figure and a Blender rendering movie can be viewed online and downloaded (for a better resolution) at https://doi.org/10.6084/m9.figshare.30535781.v2 and https://doi.org/10.6084/m9.figshare.30531185.v1.
• Figure 2: Time evolution of the 3D radiation–hydrodynamical simulation with $i=30^\circ$ (R30). Top: tilt. Middle: twist angles. Bottom: warp amplitude. Colors from blue to yellow denote increasing time. The first $100\ P_0$ ($P_0 \approx 253$ yr) are omitted to highlight the subsequent evolution. Otherwise, the twist plot would be dominated by oscillatory curves during the first $100\ P_0$. The horizontal dashed lines are the shadow's tilt (30$^\circ$) and twist angle (90$^\circ$).
• Figure 3: Time evolution of the 3D radiation–hydrodynamical simulations for $i=30^\circ$ (R30, left) and $i=90^\circ$ (R90, right). Top: surface density. Middle: vertically integrated Reynolds stress normalized by pressure. Bottom: mass accretion rate. Colors from blue to yellow denote increasing time, with $t=5$, $105$, and $505\ P_0$ highlighted by thicker lines ($P_0 \approx 253$ yr). Solid lines indicate positive values, while dashed lines indicate negative values.
• Figure 4: Slices of the radiation–hydrodynamical simulations at $r=4\ r_0$ (160 au) for $i=30^\circ$ tilted shadow (R30, left) and $i=90^\circ$ tilted shadow (R90, right) at $t=500\ P_0$ ($P_0 \approx 253$ yr). Top: density. Middle: temperature. Bottom: tangential velocity (streamlines) overlaid on the radial velocity background. Longitude lines are spaced every $60^\circ$, centered at $\phi'=\pi$; latitude lines are spaced every $30^\circ$. Although both shadows have the same intrinsic width, the polar shadow appears narrower due to projection effects. The center and edges (i.e., $\tau=1$ surfaces, at $\pm$ 0.27 of the midplane) of the shadow lanes in density and velocity plots are marked by white curves. In the density panels, the high density regions are defined as the regions within the black curves defined as the $\exp(-1/2)$ of the maximum density along the $\theta'$ direction. In the temperature panels, shock heated regions are marked by black arrows.
• Figure 5: Slices of the 3D radiation–hydrodynamical simulation with $i=30^\circ$ (R30) at $t=500\ P_0$ ($P_0 \approx 253$ yr, so $t \approx 0.13$ Myr), shown in the transformed coordinate system $(r, \theta', \phi')$, aligned with the local angular momentum vector at each radius (Figure \ref{['fig:warp_properties_evolution']}). Left to right: density, temperature, radial velocity, meridional velocity, and the deviation of the azimuthal velocity from Keplerian. All velocity components are projected into the transformed frame. Top to bottom: values at the new midplane ($\theta' = \pi/2$), at $z'/r = 0.2$ above the midplane, at the surface of maximum density, and vertical slices in the $y'-z'$ and $x'-z'$ planes. The disk rotates counter-clockwise. The arrows qualitatively indicate the azimuthal locations of velocity maxima, indicating the dominant modes that are present. Some of the grid-like patterns arise from the nearest-neighbor interpolation used in the coordinate transformation. To facilitate comparison with observations, line-of-sight velocities are presented in Figures \ref{['fig:los_zr0']} and \ref{['fig:los_zr0p2']}.