Correlation of macroscopic instability and Lyapunov times in the general three-body problem

Abstract

We conducted extensive numerical experiments of equal mass three‐body systems until they became disrupted. The system lifetimes, as a bound triple, and the Lyapunov times show a correlation similar to what has been earlier obtained for small bodies in the Solar system. Numerical integrations of several sets of differently randomized initial conditions produced the same relationship of the instability time and Lyapunov time. Marginal probability densities of the various times in the three‐body experiments are also discussed. Our high accuracy numerical method for three‐body orbit computations and Lyapunov time determinations is concisely described.

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

“…10). This contrasts with the result in Mikkola & Tanikawa (2007) for the general threebody problem where it was found that a power law with β ≈ 1.8 roughly fits 90 per cent of the data on an interval 0.94 < t l < 35.2. Since the data in Fig.…”

“…Comparing this plot to fig. 3 in Mikkola & Tanikawa (2007) we note some similarities. First, for small t l the median curve is steeper compared to larger t l values.…”

“…Now we look at the distributions of the various quantities studied so far and compare them to the results for the general three‐body problem. In the numerical experiments for the general three‐body problem in Mikkola & Tanikawa (2007), the authors fit the marginal distribution of t d (for large values of t d ) and large values of the ratio Z = t d / t l to exponential probability density functions, finding that The probability density for t l in the general three‐body problem was obtained numerically and a theoretical explanation was given in Mikkola & Tanikawa (2007) as follows. The authors assume that t d and Z are independent variables and so the marginal probability density function, ψ( t l ), can be determined via Their numerical results do not quite match up with , but the authors did fit their data with a function proportional to t −2 l .…”

“…Of the systems studied by Mikkola & Tanikawa (2007), in particular the free-fall three-body problem, the change in energy of the escaping body can vary widely depending on the interaction with the resulting binary system. This can be modelled in the Sitnikov problem by increasing the eccentricity of the binary.…”

“…An 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 authors looked for a correlation between the Lyapunov time and escape time for the planar three-body problem.…”