Mean-Motion Resonances With Interfering Density Waves

In this work, we study the dynamics of two less massive objects moving around a central massive object, which are all embedded within a thin accretion disc. In addition to the gravitational interaction between these objects, the disc-object interaction is also crucial for describing the long-term dynamics of the multi-body system, especially in the regime of mean-motion resonances. We point out that near the resonance the density waves generated by the two moving objects generally coherently interfere with each other, giving rise to extra angular momentum fluxes. The resulting backreaction on the objects is derived within the thin-disc scenario, which explicitly depends on the resonant angle and sensitively depends on the smoothing scheme used in the two-dimensional theory. We have performed hydrodynamical simulations with planets embedded within a thin accretion disc and have found qualitatively agreement on the signatures of interfering density waves by measuring the torques on the embedded objects, for the cases of $2:1$ and $3:2$ resonance. By including in interference torque and the migration torques in the evolution of a pair of planets, we show that the chance of resonance trapping depends on the sign of the interference torque. For negative interference torques the pairs are more likely located at off-resonance regimes. The negative interference torques may also explain the $1\%-2\%$ offset (for the period ratios) from the exact resonance values as observed in {\it Kepler} multi-planet systems.