# An Analytic Criterion for Turbulent Disruption of Planetary Resonances

## TL;DR

This paper derives an analytic criterion to determine when turbulent density fluctuations in protoplanetary disks can break mean motion resonances between planets, supported by numerical simulations, highlighting the importance of planet-star mass ratio.

## Contribution

The authors present a new simple analytic criterion for turbulent resonance disruption, validated by numerical integrations and N-body simulations, linking physical parameters to resonance breaking conditions.

## Key findings

- Resonance disruption depends strongly on planet-star mass ratio.
- Only low-mass planet pairs (less than about 3 Earth masses) are susceptible to turbulence-induced resonance breaking.
- Additional mechanisms are needed to explain the prevalence of non-resonant exoplanet systems.

## Abstract

Mean motion commensurabilities in multi-planet systems are an expected outcome of protoplanetary disk-driven migration, and their relative dearth in the observational data presents an important challenge to current models of planet formation and dynamical evolution. One natural mechanism that can lead to the dissolution of commensurabilities is stochastic orbital forcing, induced by turbulent density fluctuations within the nebula. While this process is qualitatively promising, the conditions under which mean motion resonances can be broken are not well understood. In this work, we derive a simple analytic criterion that elucidates the relationship among the physical parameters of the system, and find the conditions necessary to drive planets out of resonance. Subsequently, we confirm our findings with numerical integrations carried out in the perturbative regime, as well as direct N-body simulations. Our calculations suggest that turbulent resonance disruption depends most sensitively on the planet-star mass ratio. Specifically, for a disk with properties comparable to the early solar nebula with $\alpha=0.01$, only planet pairs with cumulative mass ratios smaller than $(m_1+m_2)/M\lesssim10^{-5}\sim3M_{\oplus}/M_{\odot}$ are susceptible to breaking resonance at semi-major axis of order $a\sim0.1\,$AU. Although turbulence can sometimes compromise resonant pairs, an additional mechanism (such as suppression of resonance capture probability through disk eccentricity) is required to adequately explain the largely non-resonant orbital architectures of extrasolar planetary systems.