Excitation of Orbital Eccentricities by Repeated Resonance Crossings: Requirements

Abstract

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 farther 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 resonance 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 À2 , a dimensionless alpha viscosity of 0.1, and a height-to-radius aspect ratio of 0.05 can drive two Jupiter-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 protoplanetary disk evolution, including the debris disk phase.

The dynamics of two massive planets on inclined orbits

“…Taken as a whole, we believe that these properties are consistent with a scenario in which both disk-driven migration and planet-planet interactions play critical roles in explaining the origin of extrasolar planet eccentricities (e.g. Snellgrove et al 2001;Lee and Peale 2003;Chiang 2003), although quantifying the statistical properties of the resulting planetary systems within such models remains difficult.…”

A coherent and comprehensive model of the evolution of the outer Solar System

“…However, it was also known that, if the planets cross mutual mean motion resonances during their divergent migration, their eccentricities are enhanced almost impulsively (Chiang, 2003). The eccentricity increase depends on the planetary masses and on the resonance involved.…”

Neptune Trojans as a Test Bed for Planet Formation

“…Morbidelli et al (2004) propose that in the early planetesimal disk, Jupiter Trojans are captured as Jupiter and Saturn migrate divergently across their mutual 2:1 resonance (see also Chiang [2003] for a more general discussion of divergent resonance crossings). While Jupiter and Saturn occupy the 2:1 resonance, Jupiter's 1:1 resonance is unstable; planetesimals stream through Trojan libration regions on orbits that tend to be highly inclined due to planetary perturbations.…”