On Predicting Long-Term Orbital Instability: A Relation Between the Lyapunov Time and Sudden Orbital Transitions

Attention is given to three examples representative of solar system dynamics in which the Liapunov time, TL, and the time for an orbit to make a sudden transition, TC, are found to be strongly correlated. The relation between the two times is TC varies as TL super b, with b approximately equal to 1.8. About 150 orbits were numerically integrated in the first and second examples, and about 1000 orbits in the third. In the first two examples, all three bodies were coplanar. In the third, the initial inclination of the test particle was varied from 0 to 60 deg. There was, at most, a weak dependence on inclination. The maximum departure of b from 1.8 was 14 percent, and the average departure was less than 7 percent. The correlation between TC and TL holds over at least six orders of magnitude in TC. The Liapunov time typically reaches its asymptotic value in a few thousand orbits, and then it can be used to predict sudden events that occur at much later times, e.g., the lifetime of the solar system.