ON PREDICTING LONG-TERM ORBITAL INSTABILITY - A RELATION BETWEEN THE LYAPUNOV TIME AND SUDDEN ORBITAL TRANSITIONS

In three examples representative of Solar System dynamics, we find that the Lyapunov time, T(L) (i.e., the inverse of the Lyapunov exponent) and the time for an orbit to make a sudden transition T(C) are strongly correlated. The relation between the two times is T(C) is-proportional-to T(L)b, with b congruent-to 1.8. The first example examines asteroid orbits interior to Jupiter; the sudden transition occurs when the asteroid makes a close approach to Jupiter, which occurs close to the time when the asteroid's orbit crosses Jupiter's orbit. The second example examines orbits of hypothetical asteroids between Jupiter and Saturn; the sudden transition occurs when the asteroid's orbit crosses the orbit of either of the planets. The third, considerably different, example examines massless bodies that initially orbit the smaller of the two masses of a binary system. In this case, the escape of the satellite signals the sudden transition. We have numerically integrated about 150 orbits in the first and second examples, and about 1000 orbits in the third example. In the first two examples, all three bodies were coplanar. In the third example, the initial inclination of the test particle was varied from 0-degrees to 60-degrees. There was, at most, a weak dependence on inclination. The tight clustering of the exponent b was remarkable, considering the widely different dynamical systems. The maximum departure of b from 1.8 was 14%, and the average departure was less than 7%. The correlation between T(C) and T(L) holds over at least six orders of magnitude in T(C); the longest integrations for asteroids interior to Jupiter extended for 10(8) yr. The Lyapunov time typically reaches its asymptotic value in a few thousand orbits, and then it can be used to predict sudden events (T(C)) that occur at much later times, e.g., the lifetime of the solar system.