# Long-term Hydrodynamic Simulations on the Planetesimals Trapped in the First-order Mean Motion Resonances

## TL;DR

This study uses long-term hydrodynamic simulations to analyze how asymmetric disk structures influence the eccentricity and size thresholds of planetesimals trapped in first-order mean motion resonances, revealing significant deviations from previous symmetric assumptions.

## Contribution

It provides revised analytical expressions for equilibrium eccentricity and minimum planetesimal size considering disk asymmetry effects, improving understanding of resonance trapping dynamics.

## Key findings

- Eccentricity estimates are underestimated by 30-40% for j<5 resonances.
- Asymmetric disks reduce equilibrium eccentricity and minimum size thresholds.
- Eccentricity and size effects vary with planet mass and disk asymmetry levels.

## Abstract

The resonant perturbations from planets are able to halt the drag-induced migration, and capture the inwardly drifting planetesimals into mean motion resonances. The equilibrium eccentricity of planetesimals in resonances, and the minimum size of planetesimal that can trigger resonance trapping, have been analyzed and formulated. However, the analytical works based on the assumption that the disk is axisymmetric, which is violated by the asymmetric structures developed by planets. We perform long-term 2D hydrodynamic simulations to study the dynamics of planetesimals in the $j:(j+1)$ first-order exterior resonances, and reexamine the theoretical expressions. We find the expression of equilibrium eccentricity underestimates the values for resonances with $j < 5$, in particular the 1:2 resonance that the underestimation can be $30 - 40\%$. Within the parameter space we explored, we find the equilibrium eccentricity and the minimum size are reduced in an asymmetric disk. The amount of discrepancy in eccentricity depends on the degree of asymmetric structures. For cases of Earth-sized planets, where the disk is less disturbed, the planetesimal's eccentricity can reach to the values predicted by our modified expression. For gaseous planets, however, the eccentricity can be $0.01 - 0.02$ smaller in value. We find the minimum size is 10 times smaller, and the factor seems to be independent of the planet's mass. The influences of asymmetric profiles on the eccentricity and the minimum size could affect the outcome of collisions between resonant and nonresonant planetesimals, and the amount of planetesimals migrated into the planet's feeding zone.