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Session 69 - Instabilities in Planetary Systems.
Display session, Friday, January 09
Exhibit Hall,

[69.03] A Multiple Timestep Symplectic Algorithm for Integrating Close Encounters in Planetary Systems

M. H. Lee, M. J. Duncan (Queen's U.), H. F. Levison (Southwest Research Institute)

The ability to follow close encounters between objects in planetary system integrations is important to our understanding of many of the processes that affect the structure of planetary systems. These processes include (i) the accumulation of the planets, (ii) the stability of planetary and satellite systems, and (iii) the long-term dynamics of small bodies. A class of symplectic integration algorithms known as mixed variable symplectic (MVS) integrators, which are designed specifically for planetary system integrations, has been developed in the last few years. However, the existing MVS integrators cannot handle close encounters between objects. We present a new symplectic integration algorithm that combines a variant of the MVS methods with a multiple timestep technique. The force between each pair of particles is decomposed into components that are applied with different timesteps. When the particles are well separated, only one force component is non-zero, and the algorithm has the speed of the standard MVS methods. But whenever two particles suffer a mutual encounter, the stepsize for the relevant particles is recursively subdivided to whatever level is required. The resulting method has all the desirable properties of a time-reversible symplectic method. We demonstrate the power of this method using two types of tests: (i) the few special problems with conserved quantities (e.g., the conservation of energy or the Jacobi constant), and (ii) more complex problems with published statistical results (e.g., gravitational focusing enhancement of the collision cross-section of planets and the accretion of the Moon after a giant impact). We also discuss the application of this method to the study of the late stages of planet formation.

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