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G.R. Stewart (Univ. of Colorado)
N-body simulations of the final stages of planet formation fail to form massive planets in the outer solar system unless a strong "drag" force is applied to damp the orbital eccentricites and inclinations of the largest planetesimals. A physical mechanism that could provide an effective drag is the excitation of collective gravitational waves in the disk of planetesimals. For example, Ward and Hahn (1998) have suggested that the planet Neptune could damp its orbital eccentricity by exciting apsidal waves in the Kuiper belt. I will describe a new numerical method for modeling the self-consistent dynamical interaction between a planet and an apsidal wave that can be inserted into an N-body simulation of planetary accretion. The most straight-forward method of simulating the wave would be to divide the disk into a collection of precessing wires that interact gravitationally with each other as well as with the planet. This is essentially the formalism described by Tremaine (1998) in his theory of "resonant relaxation." I have derived a more efficient method: (1) write down an infinite degree-of-freedom Hamiltonian that describes planet disk interactions; (2) expand the disk degrees-of-freedom in a truncated series of Chebyshev polynomials and reduce the Hamiltonian to a finite number of degrees of freedom; and (3) derive a new set of equations of motion for planet-disk interactions. Chebyshev polynomials are attractive because (1) they reduce the gravitational interaction of the disk to diagonal form and (2) a much smaller number of polynomials (compared to wires) is required to resolve a given wave pattern in the disk. I will present the results of a simulation using this method to model the dynamical evolution of a planet that is simultaneously excited by another planet and damped by an apsidal wave in the disk. Numerical results indicate that a planet's eccentricity shows an initial steady decay as the disk wave builds in amplitude, but at later times the planet's eccentricity exhibits slow oscillations as the wave gives back energy to the planet.