# Resonant Planet-Disk Interactions in the Solar System

Tuesday, October 30, 2001 - 2:00pm - 3:00pm

Keller 3-180

Glen Stewart (University of Colorado)

During the formation of the solar system, the planets grew from a disk of gas and dust in orbit around the newly formed sun. When the embryonic planets grew large enough, they began to resonantly excite spiral density waves and spiral bending waves in the surrounding disk of material. These planet-disk interactions resulted in substantial changes in the orbital eccentricities, inclinations and semimajor axes of the planetary embryos, and therefore likely determined the final number of planets and their orbital spacing in the solar system. The discovery of many extrasolar planets with very different orbital parameters has stimulated many recent efforts to model planet-disk interactions in large-scale fluid dynamic simulations.

Idealized mathematical models of planet-disk interactions were first developed in the context of observed satellite interactions with planetary rings. Inelastic collisions between ring particles damp their relative velocities to such low levels that pressure forces can often be neglected in Saturn's rings. Hence, satellites of Saturn launch spiral density waves in the rings that can be modeled by a self-gravitating, pressureless fluid. These models have also been applied to resonant interactions between Neptune and the Kuiper belt. The spatial dependence of the disk waves can be interpreted as the time evolution of a pendulum that is subject to a period force with a slowly varying frequency. Nonlinear extensions of the model lead to new phenomena, such as spatial autoresonance, which may help explain why strong satellite resonances open up wide gaps in Saturn's rings.

The idealized models of planet-disk interactions can be reformulated as variational principles. The variational principle allows one to derive conservation laws by Noether's method and provides a convenient starting point for deriving discrete approximations to the disk dynamics. Although lagrangian variational principles are relatively easy to derive, it is also possible to derive eulerian variational principles that may be be better suited for treating gaseous disks in the early solar system where pressure forces dominated wave propagation rather than self-gravity.

Idealized mathematical models of planet-disk interactions were first developed in the context of observed satellite interactions with planetary rings. Inelastic collisions between ring particles damp their relative velocities to such low levels that pressure forces can often be neglected in Saturn's rings. Hence, satellites of Saturn launch spiral density waves in the rings that can be modeled by a self-gravitating, pressureless fluid. These models have also been applied to resonant interactions between Neptune and the Kuiper belt. The spatial dependence of the disk waves can be interpreted as the time evolution of a pendulum that is subject to a period force with a slowly varying frequency. Nonlinear extensions of the model lead to new phenomena, such as spatial autoresonance, which may help explain why strong satellite resonances open up wide gaps in Saturn's rings.

The idealized models of planet-disk interactions can be reformulated as variational principles. The variational principle allows one to derive conservation laws by Noether's method and provides a convenient starting point for deriving discrete approximations to the disk dynamics. Although lagrangian variational principles are relatively easy to derive, it is also possible to derive eulerian variational principles that may be be better suited for treating gaseous disks in the early solar system where pressure forces dominated wave propagation rather than self-gravity.