On the relationship between instability and Lyapunov times for the three-body problem

Abstract

In this study we consider the relationship between the survival time and the Lyapunov time for three‐body systems. It is shown that the Sitnikov problem exhibits a two‐part power‐law relationship as demonstrated by Mikkola & Tanikawa for the general three‐body problem. Using an approximate Poincaré 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. 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.

The decay of triple systems

“…For long decay times (100, 000T cr < T < 1, 000, 000T cr ), this distribution is approximately a power law, f (T ) ∝ T −α , with α ≈ 1.74. This result qualitatively agrees with the results of other studies [6][7][8][9][10].…”

“…This figure also shows two fits to f (T ): an exponential function (solid curve) and a power law (dashed curve). We can see from the figure that the power-law provides a much better fit than the exponential, in agreement with the earlier studies [6][7][8][9][10]. The power-law index α = 1.74 is located within the range obtained in [6][7][8][9][10], from 1.4 to 2.2.…”

“…We accordingly expect that, on long time scales, the distribution function for the decay times should have a power-law character (and not an exponential character, like radioactive decay). This hypothesis has been verified in a number of studies (see, for example, [6][7][8][9][10]), using various methods to specify the initial conditions in the three-body problem. These studies have shown that the differential distribution of the decay times can be fit well by a power law f (T ) ∝ T −α on long time scales, with α in the range from 1.4 to 2.2.…”

Shadowing unstable orbits of the Sitnikov elliptic three-body problem

“…Urminsky & Heggie (2009) demonstrated that the Poincaré map ϕ with can be approximated by a simplectic map φ : ( t 0 , E 0 ) → (t 1 , E 1 ) where and a , b and C are constants. The quantities t 1/2 and E 1/2 are approximations of the time and energy values of m 3 at a local maximum distance from the SOS.…”

Short Lyapunov time: a method for identifying confined chaos

“…Urminsky and Heggie (2009) used the Poincaré surface of section of the three-body problem to refine the values of these constants. However, Milani and coworkers (Milani & Nobili 1992;Milani et al 1997) argued against the idea of this simple relation by drawing attention to the discovery of asteroids with short T L that do not become planet crossers, nor exhibit any large radial variations, for a thousand times the T L .…”