Eccentric discs as a gateway to giant planets outward migration

TL;DR

This paper investigates the mechanism behind inward versus outward migration in the planet-dominated Type II regime, addressing the observed correlation between migration direction and gap depth. Using long-term 2D hydrodynamic simulations with Fargo3D across $m_p=1$–$13\,M_J$ and $h=0.03$–$0.1$, the authors show that outward migration arises when the outer gap edge becomes eccentric due to the dominance of the outer 1:3 Lindblad resonance over the 1:2 resonance, and that the surface-density ratio $\Sigma_{1:2}/\Sigma_{1:3}$ tracks this transition. They demonstrate that the gap-depth parameter $K=\frac{q^{2}}{\alpha h^{5}}$ effectively separates inward from outward migration far from the transition, but does not uniquely locate the zero-torque point, and that a precise gap-width description is needed. The study extends to higher-mass planets, where a subset enters a different, highly eccentric regime, and presents a toy model that reproduces outward tracks and the stalling radius, highlighting the significant role of gap structure in dictating migration trajectories and informing population-level expectations for exoplanet systems.

Abstract

Recent studies on planet-dominated Type II migration demonstrated the presence of a correlation between the direction of planet migration and the parameter K describing the depth of the planetary gap. It was found that high (low) value for K correspond to outward (inward) migration. In this paper we aim at understanding the mechanism driving inward/outward migration and why it correlates with the gap depth. We performed a suite of 2D, live-planet, long-term simulations of massive planets migrating in discs with the hydro-code Fargo3D. We focus on a range of planet masses (1-13 m_J) and disc aspect ratios (0.03-0.1) and analyze the evolution of orbital elements and gap structure. We also study the torque contributions from outer Lindblad resonances to investigate their role in the migration outcome. We find that, while all planets initially migrate inwards, those with high enough K eventually enter a phase in which the torque reverses sign and migration becomes outwards, until eventually stalling. This behavior is associated with eccentricity growth in the outer disc and changes in the gap structure. We identify the surface density ratio at the 1:2 and 1:3 outer Lindblad resonances as a key output diagnostic that correlates with the migration direction. This ratio regulates the migration for all the cases where the massive planet remains in an almost circular orbit and the outer gap region exhibits moderate eccentricity. This characteristic sequence of inward-reversal-outwards-stalling occurs for a variety of K values and thus further work is required to identify the simulation input parameters that determine the onset of this sequence. Our results suggest that outward migration in the planet-dominated regime is primarily governed by the relative importance of the 1:2 and 1:3 resonances and, therefore, the gap profile plays a crucial role in determining the direction of migration.

Figures (14)

• Figure 1: Semi-major axis (upper panels) and eccentricity (lower panels) as a function of the evolutionary time. The left panels show the 'light' disc simulations, the right panels show the 'massive' disc simulations. As indicated in the legend, the solid/dashed/dotted lines show the orbital parameters for simulations with 1 $m_{\rm Jup}$ / 3 $m_{\rm Jup}$ / 13 $m_{\rm Jup}$ planets. The different colors correspond to different disc aspect ratios.
• Figure 2: Simulation M-m1-h05. The colormaps show the azimuthal average of the disc normalized density $\Sigma/\Sigma_0$ (left panel) and of the disc eccentricity (right panel) as a function of the disc radius (x-axis) and of evolutionary time $t/t_{\nu,0}$ (y-axis). The white line in both plots shows the planet migration track.
• Figure 3: Analysis of the migration behavior in simulation M-m1-h05. The left panel shows the semi-major axis as a function of the evolutionary time, with a zoom in the region where the planet changes its direction of migration, marked via the green dot. The other colored dots mark selected snapshots just before and after the change in migration direction. The central-left panel shows the density profiles at the selected snapshots. The central-right panel shows the disc (dark purple) and planet eccentricity (medium purple) evolution; the colored regions highlight the selected snapshots. The plot in the right panel illustrates the density ratio at the 1:2 and 1:3 resonances.
• Figure 4: Portion of the normalized torque responsible for the change in the semi-major axis as a function of the density ratio at the 1:2 and 1:3 resonances $\Sigma_{1:2}/\Sigma_{1:3}$. Different colors indicate different disc aspect ratios at 5 au, as indicated in the legend. Markers indicate the state of each simulation at $t=10\ t_{\nu}$. The lines show the evolution of each simulation, whose start is shown by the circles. The inset shows the evolution track for the simulation M-m1-h05. The high (low) line opacity corresponds to the lower (higher) disc mass regimes.
• Figure 5: Migration track (left panel) and density profiles (right panel). The colored dots in the left panel show some selected snapshots for inward (orange), outward (light blue) and stalling (purple and pink) phase of migration. The density profiles in the right panel are shown in colors corresponding to the dots in the left panels.