Excitation of Orbital Eccentricities by Repeated Resonance Crossings: Requirements

Divergent migration of planets within a viscous circumstellar disk canengender resonance crossings and dramatic excitation of orbital eccentricities.We provide quantitative criteria for the viability of this mechanism. For theorbits of two bodies to diverge, a ring of viscous material must be shepherdedbetween them. As the ring diffuses in radius by virtue of its intrinsicviscosity, the two planets are wedged further apart. The ring mass must besmaller than the planetary masses so that the crossing of an individualresonance lasts longer than the resonant libration period. At the same time,the crossing cannot be of such long duration that the disk's direct influenceon the bodies' eccentricities interferes with the resonant interaction betweenthe two planets. This last criterion is robustly satisfied because resonantwidths are typically tiny fractions of the orbital radius. We evaluate ourcriteria not only for giant planets within gaseous protoplanetary disks, butalso for shepherd moons that bracket narrow planetary rings in the solarsystem. A shepherded ring of gas orbiting at a distance of 1 AU from asolar-type star and having a surface density of less than 500 g/cm^2, adimensionless alpha viscosity of 0.1, and a height-to-radius aspect ratio of0.05 can drive two Jovian-mass planets through the 2:1 and higher-orderresonances so that their eccentricities amplify to values of several tenths.Because of the requirement that the disk mass in the vicinity of the planets besmaller than the planet masses, divergent resonance crossings may figuresignificantly into the orbital evolution of planets during the later stages ofdisk evolution, including the debris disk phase.Comment: ApJ, in press, to appear in February 2003 issu