Computer Science – Numerical Analysis
Scientific paper
May 1994
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994aj....107.1890f&link_type=abstract
The Astronomical Journal, vol. 107, no. 5, p. 1890-1899
Computer Science
Numerical Analysis
19
Asteroids, Chaos, Gaps, Jupiter (Planet), Orbital Resonances (Celestial Mechanics), Stability, Stellar Orbits, Apsides, Eccentricity, Liapunov Functions, Numerical Analysis, Numerical Integration, Precession
Scientific paper
This paper builds on our recent efforts (Soper et al., (1990); Lecar et al., (1992); Franklin et al., (1993); Murison et al., (in press)) to establish a link between orbital chaos and obvious instability by examining many hypothetical orbits and the few minor planets (only six among those numbered) at the 2:1 mean motion resonance with Jupiter. In a realistic, though planar, approximation we show that orbits of moderate-to-high chaos (i.e., those with short Lyapunov times, T(L)) are common throughout the resonance, while weakly chaotic and regular ones are remarkably rare. Less realistic approximations--the elliptic restricted problem, for example--are shown to overestimate decidedly the stability of orbits at resonance. If we adopt the empirical relation between T(L) and an escape or collision time developed in the above papers, then we find that essentially all orbits with proper eccentricities, e(prop) less than 0.15 and greater than 0.52, and the vast majority of any eccentricity with libration amplitudes, bar-phi, less than 50 deg, should be absent today, but that a modest fraction of those with bar-phi = 80 deg +/- 30 deg can survive. The six real minor planets lie in this range and five of them, we show, are clearly stable. Some, but only some, of the cases predicted to be unstable have been verified to be so by numerical integration, which adds evidence that sufficiently chaotic orbits at resonance can escape or diffuse from it. Our view of the origin of the Kirkwood gap at this resonance therefore differs from the suggestion that a gap can (only) be formed because bodies otherwise librating stably can have their eccentricities pumped to such a degree that a collision with Mars becomes possible. However, before this Kirkwood gap can be thoroughly and satisfactorily accounted for, some integrations extending closer to the solar system age are required.
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