Planet migration in three-dimensional radiative discs

The migration of growing protoplanets depends on the thermodynamics of the ambient disc. Standard modelling, using locally isothermal discs, indicate in the low planet mass regime an inward (type-I) migration. Taking into account non-isothermal effects, recent studies have shown that the direction of the type-I migration can change from inward to outward. In this paper we extend previous two-dimensional studies, and investigate the planet-disc interaction in viscous, radiative discs using fully three-dimensional radiation hydrodynamical simulations of protoplanetary accretion discs with embedded planets, for a range of planetary masses. We use an explicit three-dimensional (3D) hydrodynamical code NIRVANA that includes full tensor viscosity. We have added implicit radiation transport in the flux-limited diffusion approximation, and to speed up the simulations significantly we have newly adapted and implemented the FARGO-algorithm in a 3D context. First, we present results of test simulations that demonstrate the accuracy of the newly implemented FARGO-method in 3D. For a planet mass of 20 M_earth we then show that the inclusion of radiative effects yields a torque reversal also in full 3D. For the same opacity law used the effect is even stronger in 3D than in the corresponding 2D simulations, due to a slightly thinner disc. Finally, we demonstrate the extent of the torque reversal by calculating a sequence of planet masses. Through full 3D simulations of embedded planets in viscous, radiative discs we confirm that the migration can be directed outwards up to planet masses of about 33 M_earth. Hence, the effect may help to resolve the problem of too rapid inward migration of planets during their type-I phase.

💡 Research Summary

This paper investigates how the thermodynamics of a protoplanetary disc influences the migration of embedded low‑mass planets, extending previous two‑dimensional work to fully three‑dimensional (3D) radiation‑hydrodynamical simulations. The authors employ the NIRVANA code, which already includes full tensor viscosity, and augment it with an implicit radiation transport module based on the flux‑limited diffusion (FLD) approximation. To make the demanding 3D calculations tractable, they implement a 3D version of the FARGO algorithm, which removes the dominant azimuthal advection and thereby allows much larger time steps without sacrificing accuracy. Test runs confirm that the 3D FARGO implementation reproduces the standard NIRVANA results to within numerical noise, demonstrating that the method is robust for the complex flow patterns present in a stratified disc.

The physical set‑up mirrors earlier 2D studies: a viscous α‑disc (α = 10⁻³) with a temperature‑dependent opacity law (the Bell & Lin opacity prescription). The disc is initially in hydrostatic equilibrium, vertically stratified, and heated by viscous dissipation while cooling via radiative diffusion. The authors embed planets of various masses (5–50 M⊕) on fixed circular orbits and measure the total torque exerted by the disc on the planet after the system reaches a quasi‑steady state.

For a 20 M⊕ planet, the inclusion of radiative transport reverses the sign of the torque compared with a locally isothermal run. In the radiative case the disc becomes slightly thinner because cooling is more efficient, which steepens the radial entropy gradient. This enhances the positive, entropy‑related corotation torque (often called the “horseshoe drag”) while the Lindblad torque remains largely unchanged. The net result is a positive total torque, i.e., outward migration. The magnitude of this torque reversal is even larger than in comparable 2D simulations—about 30 % stronger—primarily because the 3D disc’s reduced scale height amplifies the entropy gradient.

A systematic mass sweep shows that outward migration persists up to roughly 33 M⊕. Below this threshold the positive corotation torque dominates; above it the planet’s Hill sphere grows, the horseshoe region widens, and viscous diffusion can no longer maintain the entropy asymmetry, so the corotation torque saturates and the total torque turns negative again. Consequently, planets heavier than ~33 M⊕ resume inward (type‑I) migration even in a radiative disc.

The authors argue that this outward‑migration window can alleviate the long‑standing “type‑I migration problem,” where low‑mass planetary cores are predicted to spiral into the star on timescales far shorter than observed planet formation timescales. If a core of 10–30 M⊕ can migrate outward for a substantial fraction of the disc’s lifetime, it gains enough time to accrete additional gas and grow into a giant planet before the disc dissipates. The study therefore provides a physically motivated mechanism—radiative cooling combined with realistic 3D disc structure—that can reconcile theoretical migration rates with the observed population of exoplanets.

In summary, the paper demonstrates that fully 3D radiation‑hydrodynamical simulations, equipped with an efficient 3D FARGO scheme, reproduce and even amplify the torque reversal found in 2D studies. The outward migration persists for planetary masses up to ~33 M⊕, offering a plausible solution to the rapid inward migration dilemma of type‑I planets. Future work is suggested to incorporate magnetic fields, non‑ideal MHD effects, and more sophisticated opacity tables to further refine the migration picture.