Indirect forces in disc-planet interaction

TL;DR

This work quantifies how indirect forces, arising from the non‑inertial star frame, alter disc–planet coupling. Using a non‑self‑gravitating, 2D disc model with adiabatic thermodynamics, the authors combine nonlinear hydrodynamic simulations in the astrocentric frame with linear theory to isolate the indirect term’s impact. They show that the indirect force mainly affects the $m=1$ perturbation for low‑mass planets, while higher harmonics become relevant at larger masses; the indirect torque on the disc exhibits oscillatory, radially growing behavior that conserves wave angular momentum only when the indirect torque is included in both the disc structure and torque calculations. For low‑mass planets, the planet’s torque is only weakly altered by including indirect forces, implying Type I migration is robust to IT neglect; for high‑mass, gap‑opening planets IT can significantly modify the net torque and migration regime. The study underscores the importance of including the indirect term for self‑consistent angular‑momentum accounting and recommends clearly reporting its inclusion in disc–planet simulations.

Abstract

Gravitational coupling between a protoplanetary disc and an embedded planet is often studied in a frame attached to a central star. This frame is non-inertial because of the stellar reflex motion, leading to indirect forces arising in the star-planet-disc system. Here we examine the impact produced by these forces on several aspects of disc-planet coupling using analytical and numerical means. We explore how neglecting indirect forces changes (1) the spatial pattern of the surface density perturbation in the disc, (2) the calculation of the torque exerted on the disc by the planet, and (3) the torque on the planet exerted by the disc. For low-mass planets, in the linear regime, the differences in the perturbation pattern are only in its $m=1$ azimuthal harmonic, with an amplitude increasing with the distance from the star. In this regime both the torque on the planet and the deposition torque density in the disc are only weakly affected by non-inclusion of indirect forces, corroborating some results of studies neglecting indirect forces altogether. For higher mass planets, a broader range of azimuthal harmonics of the perturbation are affected. Also, indirect forces have a stronger effect on the planetary torque and on planet migration in the Type II regime. We highlight the importance of including the planetary indirect force in the calculation of the torque on the disc (if disc evolution accounts for indirect force) to ensure conservation of angular momentum carried by the planet-driven density waves. The corresponding indirect torque has an oscillatory, radially-diverging character.

Figures (14)

• Figure 1: 2D map of the difference $S$ of the surface densities $\Sigma(R,\phi)$ from simulations with/without the indirect term, normalized by the local amplitude of non-axisymmetric perturbation (i.e of density wave) in a simulation with IT. Both runs use $M_\mathrm{p} = 0.01 M_\mathrm{th}$ and all other parameters are identical. Right panel is the radial profile of the azimuthal average of $|S|$ providing an idea of the relative role played by the IT.
• Figure 2: Comparison of the disc response to the planetary perturbation computed for a low planetary mass $M_\mathrm{p} = 0.01 M_\mathrm{th}$, corresponding to the linear regime of disc-planet interaction. Simulations are run for an adiabatic disc with $\gamma=1.4$, $h_\mathrm{p} = 0.1$, temperature and surface density power law indices $p=q=1$. (a) Comparison of the azimuthal profiles of the surface density perturbation $\delta\Sigma$ of the planet-driven density wave at different radii $R$ in the outer disc (normalized by the linear theory prediction $\delta\Sigma_\mathrm{lin}$, see Section \ref{['sec:analytical']}) extracted from simulations with (dashed) and without (solid) the indirect term. Only small differences are visible. (b),(c) Azimuthal Fourier components of $\delta\Sigma$ for several low values of $m$ in the inner (b) and outer (c) regions of the disc, obtained from simulations with (dashed) and without (solid) the indirect term. One can see that indirect term affects only $m=1$ component of $\delta\Sigma$ and predominantly in the outer disc, as expected from linear theory, which applies in this case because of small $M_\mathrm{p}$.
• Figure 3: Same as Fig. \ref{['fig:fourier_comparison']} but for $M_\mathrm{p} = 0.3 M_\mathrm{th}$. One can see that the differences between the calculations with the IT and without it are considerably larger than in Fig. \ref{['fig:fourier_comparison']}, no longer confined only to $m=1$ perturbation component, and noticeable also in the inner disc. The oscillatory structure of $\delta\Sigma_m$ in the outer disc is explained in Appendix \ref{['sec:fourier_oscillations']}.
• Figure 4: Illustration of the torque density computed (a,b) without the IT and (c) with the IT included. Data for the outer disc ($R>R_\mathrm{p}$) are taken from the athena++ simulations for an adiabatic disc with $h_\mathrm{p} = 0.1$, $p=q=1$ and $M_\mathrm{p} = 0.01 M_\mathrm{th}$. In panel (a), only the direct torque density $\mathrm{d} T_\mathrm{d}/\mathrm{d} R$ given by the equation (\ref{['eq:dTdRd']}) is shown; the solid blue curve is $\mathrm{d} T_\mathrm{d}/\mathrm{d} R$ computed based on $\Sigma$ from a simulation without the IT, while the dashed orange curve is the direct torque computed using $\Sigma$ from a simulation with the IT properly included. Panel (b) is a vertical zoom of panel (a) illustrating the torque wiggles Cimerman2024b. In panel (c) the $\Sigma$ data are from the simulation with the IT included, and the different curves show the full torque $\mathrm{d} T/\mathrm{d} R$ (black solid) defined by equation (\ref{['eq:dTdR']}), the direct torque $\mathrm{d} T_\mathrm{d}/\mathrm{d} R$ (orange dashed, same curve as in panels (a),(b)), and the indirect torque $\mathrm{d} T_\mathrm{id}/\mathrm{d} R$ (green dot-dashed). Note the conspicuous oscillations of growing amplitude exhibited by the indirect and total torque densities as $R$ increases in panel (c).
• Figure 5: Comparison of the full excitation torque density $\mathrm{d} T/\mathrm{d} R$, including the indirect contribution $\mathrm{d} T_\mathrm{id}/\mathrm{d} R$, obtained in our $M_\mathrm{p}=0.01M_\mathrm{th}$ simulation with athena++ (solid blue) and computed using linear calculation (dot-dashed orange). Both calculations produce oscillating $\mathrm{d} T/\mathrm{d} R$; the difference in amplitudes of oscillations is discussed in Section \ref{['sec:analytical']}.