Excitation of Orbital Eccentricities by Repeated Resonance Crossings:   Requirements
  Divergent migration of planets within a viscous circumstellar disk can engender resonance crossings and dramatic excitation of orbital eccentricities. We provide quantitative criteria for the viability of this mechanism. For the orbits of two bodies to diverge, a ring of viscous material must be shepherded between them. As the ring diffuses in radius by virtue of its intrinsic viscosity, the two planets are wedged further apart. The ring mass must be smaller than the planetary masses so that the crossing of an individual resonance lasts longer than the resonant libration period. At the same time, the crossing cannot be of such long duration that the disk's direct influence on the bodies' eccentricities interferes with the resonant interaction between the two planets. This last criterion is robustly satisfied because resonant widths are typically tiny fractions of the orbital radius. We evaluate our criteria not only for giant planets within gaseous protoplanetary disks, but also for shepherd moons that bracket narrow planetary rings in the solar system. A shepherded ring of gas orbiting at a distance of 1 AU from a solar-type star and having a surface density of less than 500 g/cm^2, a dimensionless alpha viscosity of 0.1, and a height-to-radius aspect ratio of 0.05 can drive two Jovian-mass planets through the 2:1 and higher-order resonances so that their eccentricities amplify to values of several tenths. Because of the requirement that the disk mass in the vicinity of the planets be smaller than the planet masses, divergent resonance crossings may figure significantly into the orbital evolution of planets during the later stages of disk evolution, including the debris disk phase. 
E. I. Chiang (UC Berkeley)