  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.
