**31st Annual Meeting of the DPS, October 1999**

*Session 36. Planet Formation: Solar Nebula*

Contributed Oral Parallel Session, Wednesday, October 13, 1999, 10:30am-12:00noon, Sala Kursaal
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## [36.01] A New Method for Simulating Gravitational Planet-Disk Interactions

*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.

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