Turbulence in Extrasolar Planetary Systems Implies that Mean Motion Resonances are Rare

This paper considers the effects of turbulence on mean motion resonances in extrasolar planetary systems and predicts that systems rarely survive in a resonant configuration. A growing number of systems are reported to be in resonance, which is thought to arise from the planet migration process. If planets are brought together and moved inward through torques produced by circumstellar disks, then disk turbulence can act to prevent planets from staying in a resonant configuration. This paper studies this process through numerical simulations and via analytic model equations, where both approaches include stochastic forcing terms due to turbulence. We explore how the amplitude and forcing time intervals of the turbulence affect the maintenance of mean motion resonances. If turbulence is common in circumstellar disks during the epoch of planet migration, with the amplitudes indicated by current MHD simulations, then planetary systems that remain deep in mean motion resonance are predicted to be rare. More specifically, the fraction of resonant systems that survive over a typical disk lifetime of 1 Myr is of order 0.01. If mean motion resonances are found to be common, their existence would place tight constraints on the amplitude and duty cycle of turbulent fluctuations in circumstellar disks. These results can be combined by expressing the expected fraction of surviving resonant systems in the approximate form P_b = C / N_{orb}^{1/2}, where the dimensionless parameter C = 10 - 50 and where N_{orb} is the number of orbits for which turbulence is active.