Outward Migration of a Gas Accreting Planet: A Semi-Analytical Formula

TL;DR

This work explains the observed pile-up of gas giants beyond 1 au by identifying outward migration driven by azimuthal corotation torque arising from planetary gas accretion. It builds a semi-analytical framework that combines gap-depth physics with accretion feedback, yielding a general type II (full) migration formula that unifies outward and inward migration across a broad $(q,\alpha,h)$ space. The key result is the outward-migration condition $K'_{acc} \lesssim K' \lesssim K'_{gap}$ with $K'_{acc} \approx 0.03$ and $K'_{gap} \approx 50$, and a log-linear interpolation to connect to standard gap+dep regimes, reproducing simulation results within a factor of ~2. Concurrent growth and migration show outward migration can retain gas giants at large radii, potentially explaining the exoplanet pile-up and guiding population synthesis for quantitative comparisons.

Abstract

Type II orbital migration is a key process to regulate the mass and semimajor axis distribution of exoplanetary giant planets. The conventional formula of type II migration generally predicts too rapid inward migration to reconcile with the observed pile-up of gas giant beyond 1 au. Analyzing the recent high-resolution hydrodynamical simulations by Li et al. (2024) and Pan et al. (2025) that show robust outward migration of a gas accreting planet, we here clarify the condition for the outward migration to occur and derive a general semi-analytical formula that can be applied for broad range of planet mass and disk conditions. The striking outward migration is caused by azimuthal asymmetry in corotation torque exerted from cicumplanetary disk regions (connecting to horseshoe flow) that is produced by the planetary gas accretion, while the conventional inward migration model is based on radial asymmetry in the torques from the circumstellar protoplanetry disk. We found that the azimuthal asymmetry dominates and the migration is outward, when the gap depth defined by the surface density reduction factor of $1/(1+K')$ is in the range of $0.03 \lesssim K' \lesssim 50$. Using simple models with the new formula, we demonstrate that the outward migration plays an important role in shaping the mass and semimajor axis distribution of gas giants. The concurrent dependence of planets' accretion rate and migration direction on their masses and disk properties potentially reproduces the observed pile-up of exoplanetary gas giants beyond 1 au, although more detailed planet population synthesis calculations are needed in the future.

Figures (5)

• Figure 1: The efficiency of gas accretion onto planet as a function of $q_{\rm th}$, predicted by Li/Choksi (red lines) and TT (blue lines) formulas. In panels (a), the ability of planetary gas accretion ($\xi = \dot{m}_{\rm p,loc}/\dot{m}_{\rm d}$) are plotted. The dependence on $\alpha$ is shown in the upper, middle and lower panels, with a fixed value of $h = 0.05$. In panels (b), the actual ability with a cap of the disk accretion ($f_{\rm p}=\dot{m}_{\rm p}/\dot{m}_{\rm d}$) is shown. The panels (c) show the corresponding reduction $f_{\rm pass} (=1-f_{\rm p})$ in the global gas surface density.
• Figure 2: The $\Sigma_0$ distribution (the upper panel) and gas accretion rate through the disk (the lower panel) in the steady state with the effect of gas accretion onto the planet and host star (sinks) is plotted in the case of $\dot{m}_{\rm d}=10^{-8} M_\odot/\rm y$, $\alpha = 10^{-3}$, $h = 0.05$, $r_{\rm in}=0.03\, {\rm au}$, and $r_{\rm p}=1\, {\rm au}$. The green, blue, and skyblue lines are the distribution without any sink, and those with the planet sinks with $\xi=2$$(f_{\rm p} = 2/3)$ and $\xi=10$$(f_{\rm p} = 0.91)$, respectively. For $\alpha = 10^{-3}$ and $h = 0.05$, $\xi=2$ corresponds to $q = 1.6 \times 10^{-5}$ and $q = 1.86 \times 10^{-3}$, and $\xi=10$ corresponds to $q = 4.2 \times 10^{-5}$ and $q = 4.9 \times 10^{-4}$. With the sinks, $\dot{m}_{\rm d}$ decreases by the accretion onto the planet inside $1\, {\rm au}$ and vanishes by the accretion onto the host star at $r_{\rm in}$.
• Figure 3: The migration rates obtained by Pan2025 and Li2024 (upper panels). The scaling factor is $\dot{m}_{\rm d}/M_*$, where $\dot{m}_{\rm d}$ is the disk mass accretion rate and $M_*$ is the host star mass. The simulation parameters are the planet to star mass ratio $q=m_{\rm p}/M_*$, the disk viscosity parameter $\alpha$, and the disk aspect ratio $h = H/r$. Except for panel (d), $h = 0.05$ is used. In panel (d), $\alpha = 0.04$. The vertical scales are different in panel (b) for visual convenience. In each of the lower panels, $K'$ corresponds to the parameters in the directly upper panel. The outward migration is predicted in $K'_{\rm acc} < K' < K'_{\rm gap}$ for $\xi_{\rm acc} = 2$ ($K'_{\rm acc} = 0.03$) and $K'_{\rm gap}=50$.
• Figure 4: The general semi-analytical migration formula [type II (full) formula] including the azimuthal asymmetry in addition to the effects of the gap opening and the global depletion due to the planetary gas accretion (green lines). It is compared with the type II (gap) formula with only the gap opening effect (magenta lines), the type II (gap+dep) formula with the gap opening and the global depletion effects (blue lines), and four systematic parameters survey results of hydrodynamical simulations by Pan2025 (brown squares), which are the same panels as in the upper panels in Fig. \ref{['fig:mig']}.
• Figure 5: The mass and semimajor axis evolution of independent planets starting from $m_{\rm p} = 10 M_\oplus$ at 9 different locations of $a = 1$ - 30 au (Panel (a)). For the disk, the self-similar solution with $\dot{m}_{\rm d} = 3 \times 10^{-8} (1 + t/t_{\rm diff})^{-3/2} M_\odot/{\rm yr}$ with $t_{\rm diff}=10^6\, \rm yrs$ and $T = 130\, (r/{1 \,\rm au})^{-1/2}\rm K$ is used. The evolution with type II (full) formula in Section \ref{['sec:general']} is shown in the green lines. For comparison, the evolutions with type II (gap) and type II (gap+dep) formulas are also shown in the magenta and blue lines. The evolution is integrated until $t=10^7$ yrs. In Panels (b), (c), and (d), time evolution of the semimajor axis, the mass, and the gap parameter ($K'$) of individual planets are shown with type II (full) formula. In Panel (e), the result with enhanced Hill accretion rate by a factor 10 in type II (full) formula (dark green) is compared to the fiducial result (green).