How two-dimensional are planet-disc interactions? II. Radiation hydrodynamics and suitable cooling prescriptions

TL;DR

The paper assesses whether 2D radiation-hydrodynamic models can faithfully reproduce gap-opening dynamics observed in 3D disks across thermodynamic regimes, by comparing locally isothermal, adiabatic, local $\beta$ cooling, and fully radiative treatments. It develops effective cooling timescales $\beta^{3D}$ and $\beta^{2D}$ to translate full radiative behavior into tractable parameterizations and demonstrates that 2D models capture the essential physics of gap opening with accuracy comparable to 3D, even when radiative diffusion is included. A key finding is that gap-opening efficiency peaks near $\beta\sim1$, and that local $\beta$ cooling overestimates radiative damping relative to fully radiative models, though 2D and 3D agree on overall trends and gap morphologies. The results advocate for including radiative processes in planet–disk interaction studies, using simplified yet physically motivated cooling prescriptions to enable large parameter surveys, especially for ALMA-resolved substructures at $>10$ au.

Abstract

The ring and gap structures found in observed protoplanetary disks are often attributed to embedded gap-opening planets and typically modeled with simplified thermodynamics in the 2D, thin disk approximation. However, it has been shown that radiative cooling and meridional processes play key roles in planet-disk interaction, though their computational cost has limited their exploration. We investigate the differences between 2D and 3D models of gap-opening planets while also comparing thermodynamical frameworks ranging from locally isothermal to fully radiative. We also compare simplified cooling recipes to fully radiative models in an effort to motivate the inclusion of radiative effects in future modeling even in a parametrized manner. We perform hydrodynamical simulations in both 2D and 3D, and then compare the angular momentum deposition by planetary spirals to assess gap opening efficiency. We repeat comparisons with different thermodynamical treatments: locally isothermal, adiabatic, local beta cooling, and fully radiative including radiative diffusion. We find that 2D models are able to capture the essential physics of gap opening with remarkable accuracy, even when including full radiation transport in both cases. Simple cooling prescriptions can capture the trends found in fully radiative models, albeit slightly overestimating gap opening efficiency near the planet. Inherently 3D effects such as vertical flows that cannot be captured in 2D can explain the differences between the two approaches, but do not impact gap opening significantly. Our findings encourage the use of models that include radiative processes in the study of planet-disk interaction, even with simplified yet physically motivated cooling prescriptions in lieu of full radiation transport. This is particularly important in the context of substructure-inducing planets in the ALMA-sensitive disk regions (>10 au).

Figures (9)

• Figure 1: Snapshot of the perturbed surface density after 10 planetary orbits for our locally isothermal 3D model. Left: the entire simulation domain. Spiral shocks are weaker near either radial end of the disk due to wave damping. Top right: a zoom-in into the orange box on the left, highlighting the region where gap opening is driven. Bottom right: the angular momentum deposition function $\partial F_\mathrm{dep}/\partial R$ and the surface density evolution rate $\partial\Sigma/\partial t$ in arbitrary units computed according to cordwell-rafikov-2024. Saturated and faded colors indicate smoothed or raw data, respectively. Dashed and dotted vertical lines mark the isothermal shock distance $x_\mathrm{sh}$ (Eq. \ref{['eq:xshock']} with $\gamma=1$) and the planet's corotating region (Eq. \ref{['eq:xhorse']} with $\gamma=1$), respectively.
• Figure 2: Angular momentum deposition $\partial F_\mathrm{dep}/\partial R$ for our fiducial 2D and 3D models (dashed and solid lines, respectively) and for four different equations of state (EOS). Models with cooling ($\beta$, radiative) have $\beta^\mathrm{2D}\approx2$. As in Fig. \ref{['fig:gap-view']}, faded and saturated colors indicate raw and smoothed data, respectively, showing that our smoothing procedure has negligible effects beyond the corotating region. The bottom panel compares the four EOS for our 3D models, highlighting the role of cooling.
• Figure 3: Estimates of the perturbed surface density expected after 100 planetary orbits for the models featured in Fig. \ref{['fig:fdep-eos']}cordwell-rafikov-2024. The adiabatic and isothermal models show a double-trough gap structure, while radiative models show a single, deep gap. An apparent third dip in the adiabatic model at $R=R_\mathrm{p}$ is related to the perturbations driven by buoyancy modes near the edge of the planet's horseshoe region.
• Figure 4: The angular momentum deposition function $\partial F_\mathrm{dep}/\partial R$ for our 3D (top) and 2D models (bottom) as a function of the cooling timescale defined through Eqs. \ref{['eq:beta-3D']} & \ref{['eq:beta-2D']} for 3D and 2D, respectively. The quantity approaches zero within $x_\mathrm{sh}$ (dashed vertical lines) for very short or very long cooling timescales (blue and red lines, respectively), but peaks around $\beta\sim1$ (gray dots).
• Figure 5: Estimated gap structures after 100 planetary orbits similar to Fig. \ref{['fig:sdot-eos']} as a function of $\beta$. The gap structure transitions from double- to single-trough as $\beta$ increases towards $\beta\sim1$, and then back to a double-trough gap for $\beta\gg1$. A secondary gap is also visible for $R\sim0.6\,R_\mathrm{p}$ for $\beta\ll1$ and $\beta\gg1$, induced by the planet's secondary inner spiral arm.