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Symplectic methods for the simulation of Hamiltonian systems
by
John Butcher
The University of Auckland
For the faithful simulation of gravitational and other Hamiltonian problems, symplectic (or canonical) numerical methods have a crucial role. A typical method of this type is the 2-stage implicit Runge-Kutta method based on Gaussian quadrature. While preserving quadratic invariants, this method has the disadvantage of being fully implicit and therefore expensive to implement. In comparison, a general linear method with similar accuracy exists, which is diagonally implicit and therefore less expensive. Although it is only G-symplectic it preserves many invariants for millions of time steps and apparently forever.
Date received: October 30, 2008
Resonance, chaos and stability in the general three-body problem
by
Rosemary Mardling
School of Mathematical Sciences, Monash University
The quest to understand and predict the stability of general three-body configurations has existed since Newton formulated his equations of motion and his law of gravity. Poincaré modernized the quest with his prize-winning work in the early twentieth century, effectively inventing the theory of chaos in the process. Interest waned until the work of Kolmogorov, Arnold and Moser in the 1950s and 60s when the famous KAM tori entered the mathematical lexicon, although nothing specific could be said about three-body stability until the work of Wisdom in 1980 who studied the planar circular restricted problem (one body is massless and the other two have small mass ratio).
Here we present a new formulation which uses the chaos concept of resonance overlap to determine stability in the general three-body problem. It involves no free parameters and is entirely general and robust, with no restrictions on the masses or orbital elements.
Paper reference: http://adsabs.harvard.edu/abs/2008LNP...760...59M
Date received: November 2, 2008
Jupiter: shield or sniper?
by
Philip Sharp
University of Auckland
Coauthors: K. R. Grazier, W. I. Newman
After the formation of the planets in the solar system, a large number of small bodies orbiting the Sun remained. Over the following half a billion years, most of the larger bodies inside the orbit of Saturn were removed. The obvious mechanisms for removal are accretion by a planet, ejection from the solar system or incorporation into the clouds or belts of objects beyond Neptune.
We simulated the trajectories of four sets of 10, 000 massless particles over 100 million years using a simulation method that achieved the theoretical lower bound on the integration error. The particles in the first set were initially between Jupiter and Saturn, those in the second between Saturn and Uranus, the third between Uranus and Neptune and the fourth in the Kuiper Belt.
We found, contrary to previous analysis, that Jupiter does not protect the terrestrial planets from bombardment by bodies orbiting the Sun.
Date received: October 28, 2008
Binary star scattering encounters resulting in single stars
by
Winston Sweatman
Massey University, Albany
Binary stars can be separated into their component stars through interaction with other binary or single stars. It is important to understand this process as the dynamics of much larger stellar N-body systems can be driven by few-body encounters.
Approximating stars by point masses, there exist theoretical approaches for this process at extremes of high and low energy. These have been used to estimate the ionisation cross-section, which is the measure of likelihood of separation into single stars during an encounter.
Date received: October 21, 2008