A Comoving Framework for Planet Migration

TL;DR

A comoving coordinate framework is developed to study planet migration in protoplanetary disks, addressing the high computational cost and non-conservative issues of fixed-grid and remapping approaches. By transforming space and time with the planet’s instantaneous semi-major axis, the planet remains stationary on a fixed mesh, while an inertial source term accounts for non-inertial effects, enabling self-consistent simulations with significantly reduced cost. The method is implemented in FARGO3D and validated against conventional fixed-grid results through a benchmark showing excellent agreement until boundary effects appear on the inertial frame; the comoving approach achieves an efficiency gain of over an order of magnitude. This framework also supports efficient high-resolution studies of eccentric or multi-planet systems and paves the way toward long-term population-synthesis studies grounded in full hydrodynamics. Overall, the work provides both a practical numerical tool and a theoretical lens for analyzing planet-disk interactions with clearer physical insight and greater computational reach.

Abstract

The migration of planets within their nascent protoplanetary disks is a fundamental process that shapes the final architecture of planetary systems. However, studying this phenomenon through direct hydrodynamical simulations is computationally demanding, with traditional methods on fixed grids being ill-suited for tracking planet migration over long timescales due to their high cost and limited domain. In this work, we present a self-consistent comoving framework designed to overcome these challenges. Our method employs a coordinate transformation that scales with the planet's evolving semi-major axis, keeping the planet stationary with respect to its local computational grid. This transforms the standard hydrodynamic equations by introducing a source term that accounts for the inertial forces of the non-inertial reference frame. We implement this framework in the FARGO3D code and validate it through a benchmark test, demonstrating excellent agreement with conventional fixed-grid simulations until the latter are compromised by boundary effects. Our analysis shows that the comoving method can be over an order of magnitude more computationally efficient, dramatically reducing the cost of simulating migrating planets. Furthermore, the framework's adaptability enables efficient, high-resolution studies of planets on eccentric orbits by keeping them stationary within the computational grid. This framework serves as both a powerful numerical and theoretical tool, simplifying the time-dependent flow around a migrating planet that offers clearer physical insight. It enables long-term, self-consistent studies of planet-disk interaction, representing a crucial step towards performing planet-population synthesis based on full hydrodynamical simulations.

Figures (4)

• Figure 1: Results of the benchmark test comparing the planet migration trajectories from the comoving (blue) and inertial frame (red) simulations. In all panels, solid lines represent the planet's radial position, dashed lines show the boundaries of the respective computational domains, and the shaded regions indicate the co-orbital ring within which gravitational torques are calculated. The left panel shows the planet's radial position $r_{\rm p}$ versus comoving time $t'$. The linear decay on the log-linear plot indicates a constant migration rate in the comoving frame, consistent with the expected steady-state solution. The middle panel shows the planet's radial position versus time $t$. The agreement between the two methods is excellent until the planet in the inertial frame simulation approaches the inner grid boundary, causing its migration to stall. The right panel shows the non-linear transformation between time and comoving time given by Eq. \ref{['eq:tprime']}, which depends on the planet trajectory.
• Figure 2: Snapshots of the gas surface density, $\Sigma(r)$, in physical coordinates, comparing the fixed inertial frame simulation (top row) with the comoving frame simulation (bottom row). The surface density is plotted against the radius, $r$, and the azimuthal angle relative to the planet, $\varphi - \varphi_{\rm p}$. For direct comparison, the comoving simulation is plotted on the same physical domain as the fixed-grid one. Columns correspond to three different times. The surface density is shown using a logarithmic scale with a linear color map over the range -5.0 (darker color) to -2.5 (lighter color). The figure illustrates the contraction of the comoving frame's domain in physical space as it follows the planet inward.
• Figure 3: Snapshots of the gas surface density in comoving coordinates, comparing the fixed inertial frame (top row) and the comoving frame (bottom row) simulations at the same times as in Fig. \ref{['fig:fixed_frame']}. The surface density is plotted against the comoving radius, $r'$, and the azimuthal angle relative to the planet, $\varphi - \varphi_{\rm p}$. The surface density is shown using a logarithmic scale with a linear color map over the range -5.0 (darker color) to -2.5 (lighter color). The excellent agreement between the disk structures shows that the comoving framework accurately reproduces the local physics of planet-disk interaction of migrating planets. As the planet migrates inward, the background surface density within the comoving frame correctly increases, consistent with the initial density profile.
• Figure 4: Comparison of the gas flow streamlines around a migrating planet, shown in the comoving frame from a comoving simulation (left) and the inertial frame from a fixed grid simulation (right) for a single time snapshot ($t=3.3\times10^3)$. The left panel displays the streamlines of the dimensionless comoving velocity, ${\bf u}'$ in comoving coordinates. The right panel shows the corresponding streamlines of the inertial velocity, ${\bf v}$, in physical coordinates. The withe arrows in both panels point towards the instantaneous direction of the flow. Because the comoving frame allows the system to reach a steady state, the streamlines of ${\bf u}'$ also represent the true trajectories of fluid elements. In the time-dependent inertial frame, the streamlines of ${\bf v}$ are an instantaneous representation of the velocity field and do not correspond to particle paths. The background color map in both panels shows the gas surface density on a logarithmic scale following Figs. \ref{['fig:fixed_frame']} and \ref{['fig:comoving_frame']}.