Other
Scientific paper
May 2009
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2009dda....40.0611z&link_type=abstract
American Astronomical Society, DDA meeting #40, #6.11; Bulletin of the American Astronomical Society, Vol. 41, p.901
Other
Scientific paper
The circular restricted three-body problem (CRTBP) is a celebrated problem in which Poincare first derived the existence of chaos. In chaotic systems, nearby trajectories diverge exponentially away from each other. This implies that numerical errors become exponentially magnified, degrading the reliability of numerical solutions.
Confidence in the reliability of numerical solutions can be increased using shadowing. A shadow is an exact trajectory that stays close to a numerical one for a long time, validating the dynamics observed in the numerical solution. To find shadows, we use a refinement algorithm similar a Newton's method (Hayes and Jackson 2007).
Murison (1989) used an extremely accurate, high cost integrator to observe fractal structure in a capture-time diagram in the CRTBP. We can reproduce this structure using much cheaper integrations, and demonstrate that most of the cheap trajectories are shadowed.
We also observe a negative correlation between capture time and shadow length. We surmise that this is because the dynamical time for orbits around the smaller massive particle in the CRTBP is much shorter than dynamical time around the larger (eg., the Moon's orbital period around the Earth is much shorter than the Earth's orbital period around the Sun), and it has been observed(Hayes 2003) that shadow lengths tend to scale linearly with dynamical times. However, in this study we also found many orbits with very long capture time (Moon-like orbits) which were shadowable for long periods, indicating that such stable orbits were reliably integrated. Thus, we are able to verify the reliability of most short capture time orbits, but are unable to verify the reliability of a significant proportion of long-capture-time orbits.
Hayes Wayne B.
Zhu Yong-Kang
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