Mean Motion Resonances in Extrasolar Planetary Systems with Turbulence,   Interactions, and Damping
  This paper continues previous work on the effects of turbulence on mean motion resonances in extrasolar planetary systems. Turbulence is expected to arise in the disks that form planets, and these fluctuations act to compromise resonant configurations. This paper extends previous work by considering how interactions between the planets and possible damping effects imposed by the disk affect the outcomes. These physical processes are studied using three approaches: numerical integrations of the 3-body problem with additional forcing due to turbulence, model equations that reduce the problem to stochastically driven oscillators, and Fokker-Planck equations that describe the time evolution of an ensemble of systems. With this combined approach, we elucidate the physics of how turbulence can remove extrasolar planetary systems from mean motion resonance. As expected, systems with sufficiently large damping (dissipation) can maintain resonance, in spite of turbulent forcing. In the absence of strong damping, ensembles of these systems exhibit two regimes of behavior, where the fraction of the bound states decreases as a power-law or as an exponential. Both types of behavior can be understood through the model developed herein. For systems with weak interactions between planets, the model reduces to a stochastic pendulum, and the fraction of bound states decreases as a power-law. For highly interactive systems, the dynamics are more complicated and the fraction of bound states decreases exponentially. We show how planetary interactions lead to drift terms in the Fokker-Planck equation and account for this exponential behavior. In addition to clarifying the physical processes involved, this paper strengthens the finding that turbulence implies that mean motions resonances should be rare. 
Daniel Lecoanet, Fred C. Adams, and Anthony M. Bloch