The authors looked for a correlation between the Lyapunov time and escape time for the planar three-body problem. They suggested that for orbits in the Nekhoroshev regime, the relationship between t d and t l should be exponential i.e.Īn investigation of the relationship between escape times and Lyapunov times for the general three-body problem has recently been conducted by Mikkola Tanikawa (2007).
The dynamics of the outer asteroid belt are not controlled by a single critical KAM curve, and the authors argued that this means that there is no reason to expect a simple scaling between the Lyapunov time and the escape time.Īnother study Morbidelli Froeschl (1996) gave a two-part relationship. Murray Holman (1997) found that the relationship does not hold for some bodies in the outer belt. In another study, Murison, Lecar Franklin (1994) considered the restricted elliptic three-body problem with Jupiter as the secondary mass.Īgain, they found that the relationship (1) holds with 1.74 0.03.ĭespite all the support of the relationship (1), there have been some disagreements with the relationship. They found that (1) holds but with a slightly higher exponent value 1.9. Levison Duncan (1993) in a study of EdgeworthKuiper belt objects showed a relationship between the Lyapunov time and the time it takes for these object to cross the orbit of Neptune. There have been many other investigations into the relationship between Lyapunov times and survival times. In this example, t d was taken to be the time it took for m 3 to escape via one of the collinear Lagrange points.Ĭorrelating data from 1000 orbits, the study found that (1) holds for 1.8. In support of the relationship (1), Lecar et al. In both studies it was found that t d and t l are related by 1 where is a constant, C is a normalization constant and A is a constant of proportionality.
The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general three-body problem.įrom a study of orbits of asteroids ( Soper, Franklin Lecar 1990 ) between Jupiter and Saturn, the authors noted a relationship between the Lyapunov time, t l, and the time, t d, which an asteroid takes to cross the orbit of Jupiter or Saturn.Ī correlation between these two time-scales was also noted for asteroids in the outer asteroid belt in Lecar, Franklin Soper (1992b). Using an approximate Poincar map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the three-body problem. Using the approximate Poincar map we delineate a region of initial conditions which escape quickly orbits for these initial conditions are used to construct a new functional relationship between t l and t d.
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