How two-dimensional are planet–disc interactions?

II. Radiation hydrodynamics and suitable cooling prescriptions

¹ Ludwig-Maximilians-Universität München, Universitäts-Sternwarte, Scheinerstr. 1, 81679 München, Germany
² Astronomy Unit, Dept. of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK
³ Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
⁴ DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
⁵ Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

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

Received: 24 September 2025
Accepted: 19 November 2025

Abstract

Context. The ring and gap structures found in observed protoplanetary disks are often attributed to embedded gap-opening planets and typically modeled with simplified thermodynamics and in the 2D, thin disk approximation. At the same time, both analytical and numerical studies have shown that radiative cooling and meridional processes play key roles in planet–disk interaction, though their computational cost and modeling complexity have limited their exploration.

Aims. We investigate the differences between 2D and 3D models of gap-opening planets while also comparing different thermodynamical frameworks ranging from locally isothermal to radiative hydrodynamics. 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.

Methods. We performed high-resolution hydrodynamical simulations in both 2D and 3D, and then compared the angular momentum deposition by planetary spirals between the two sets of models to assess gap-opening efficiency. We repeated the comparisons with different thermodynamical treatments: locally isothermal, adiabatic, local β cooling, and fully radiative including radiative diffusion.

Results. 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 while also 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 they do not impact gap opening significantly.

Conclusions. Our findings encourage the use of models that include radiative processes in the study of planet–disk interaction, even with the simplified yet physically motivated cooling prescriptions that we provide in lieu of full radiation transport. This is particularly important in the context of substructure-inducing planets in the disk regions where instruments such as ALMA are sensitive (≳ 10 au).