Brownian motion in planetary migration

A residual planetesimal disk of mass 10-100M(circle plus) remained in the outer solar system following the birth of the giant planets, as implied by the existence of the Oort cloud, coagulation requirements for Pluto, and inefficiencies in planet formation. Upon gravitationally scattering planetesimal debris, planets migrate. Orbital migration can lead to resonance capture, as evidenced here in the Kuiper and asteroid belts, and abroad in extrasolar systems. Finite sizes of plane-tesimals render migration stochastic ("noisy''). At fixed disk mass, larger (fewer) planetesimals generate more noise. Extreme noise defeats resonance capture. We employ order-of-magnitude physics to construct an analytic theory for how a planet's orbital semimajor axis fluctuates in response to random planetesimal scatterings. The degree of stochasticity depends not only on the sizes of planetesimals, but also on their orbital elements. We identify the conditions under which the planet's migration is maximally noisy. To retain a body in resonance, the planet's semimajor axis must not random walk a distance greater than the resonant libration width. We translate this criterion into an analytic formula for the retention efficiency of the resonance as a function of system parameters, including planetesimal size. We verify our results with tailored numerical simulations. Application of our theory reveals that capture of resonant Kuiper Belt objects by a migrating Neptune remains effective if the bulk of the primordial disk was locked in bodies having sizes << O(100) km and if the fraction of disk mass in objects with sizes greater than or similar to 1000 km was less than a few percent. Coagulation simulations produce a size distribution of primordial planetesimals, which easily satisfies these constraints. We conclude that stochasticity did not interfere with nor modify in any substantive way Neptune's ability to capture and retain resonant Kuiper Belt objects during its migration.