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

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).

💡 Research Summary

This paper addresses a central question in the theory of planet‑disk interaction: can the widely used two‑dimensional (2D) thin‑disk approximation capture the essential physics of gap opening when realistic radiative processes are taken into account, or are fully three‑dimensional (3D) radiation‑hydrodynamic simulations indispensable? To answer this, the authors perform a systematic suite of high‑resolution hydrodynamic simulations using the PLUTO code, exploring both 2D cylindrical (R, φ) and 3D spherical (r, θ, φ) geometries. The central star has solar mass and luminosity, the disk is inviscid, and the planet’s mass is set to a modest 0.1 MJup at a reference orbital radius Rp. The computational domain extends from 0.4 Rp to 2.5 Rp, with a logarithmic radial grid, full azimuthal coverage, and a vertical extent of three scale heights in the 3D runs. Resolutions of 1200 × 4096 cells (2D) and 600 × 48 × 2048 cells (3D) provide 32 and 16 cells per scale height, respectively, ensuring that the spiral shocks and gap edges are well resolved.

Four thermodynamic treatments are examined: (i) locally isothermal (instantaneous temperature relaxation), (ii) adiabatic (no cooling), (iii) a parametrised “β‑cooling” model in which the internal energy relaxes toward a prescribed background on a timescale t_cool = β ΩK⁻¹ with a prefactor fβ≈¼, and (iv) fully radiative radiation‑hydrodynamics using the flux‑limited diffusion (FLD) approximation. In the radiative runs, the radiation energy density E_rad and flux F are evolved together with the gas, and the coupling term Qrad = −κP ρ c (aR T⁴ − E_rad) appears in the gas energy equation. Surface cooling (Qsurf) and in‑plane diffusion (Qmid) are explicitly modelled in the 2D version of the radiative scheme, together with a simple irradiation term based on stellar luminosity.

A key contribution of the work is the derivation of an effective cooling time that can be used in 2D simulations to mimic the full 3D radiative behaviour. Starting from the 3D diffusion‑limited cooling time β3D = ΩK η