Lyapunov Times of the Planet Crossing Asteroids

Arthur L. Whipple (University of Texas at Austin, Austin, Texas USA)

The maximum Lyapunov characteristic exponent, or equivalently, the 
corresponding Lyapunov time, is a measure of the exponential divergence 
of neighboring orbits in phase space.  The orbits of asteroids all have 
Lyapunov times that are short relative to the orbits of the planets.  
The planet crossing asteroids have extraordinarily short Lyapunov times, 
virtually all are less than 5000 years.

Lyapunov times for the 175 numbered asteroids whose osculating 
perihelion distances were 1.6 AU or less at JDT 2449200.5 (1 August 
1993) have been calculated by numerical integration of the orbits and 
the associated state transition matrices within a realistic three-
dimensional restricted n-body problem.  Thirty four of these asteroids 
were found to have Lyapunov times that were less than 100 years and, of 
those, 18 were less than 50 years.  Lyapunov times were found to be 
exponentially related to perihelion distance in this part of the solar 
system.

This extreme degree of chaos is due primarily to frequent close 
approaches to the terrestrial planets.  The complicated resonance 
structure of the inner solar system also contributes to the chaos.  The 
structure, causes, and implications of this chaos will be discussed.