## Recommended Citation

Babbs, Charles F., "Lifetimes of stable solutions to the planar three-body problem, revealed in patterns of fractal art" (2024). *Weldon School of Biomedical Engineering Faculty Working Papers.* Paper 37.

https://docs.lib.purdue.edu/bmewp/37

## Date of this Version

10-13-2024

## Keywords

acceleration, collision, ejection, escape, gravity, image, mass, Newton, orbit, planar, scale, self-similar, simulation, statistical theory, trajectory, Valtonen

## Abstract

This preliminary study explores statistical prediction of the lifetimes of apparently stable orbits of three particle masses, mutually attracted by gravity in the planar three body problem. Most orbits eventually break up, commonly in an end state with two of the bodies orbiting each other and moving together in one direction, with the third body moving in the opposite direction. Studies of such escape events led to a novel hypothesis, emerging from the intersection of probability, statistics, chaos, and fractal art. Statistical estimates of a constant probability of breakup per unit time allow prediction of the times of breakup of three body systems and can be applied to the open question of whether there are some three-body systems whose lifetimes are infinite. Initial conditions for the planar three body problem can be represented in a multi-dimensional feature space. If all three bodies have zero velocity at time zero and the initial positions of two of the bodies remain constant, variable initial positions of the third body in the plane can be defined in a two-dimensional subspace. In numerical simulations, classical Newtonian accelerations, speeds, and positions of all three bodies are calculated as functions of time. Maps of the plane of initial conditions indicating stable systems (in black) vs. broken systems with escape of one or more bodies unstable (in white) resemble fractals. Presented are a framework and sample calculations to test the hypothesis that there is a constant local probability of breakup per unit of time. This probability is estimated after a fixed simulation duration by sampling the measured proportion of points with stable orbits in several arbitrarily small windows surrounding a particular indexed point in the fractal plane. The measured proportions of stable systems, f(n), vary as a function of the number of time steps, n , such that for a range of window sizes, their mean value f̅(n) (1−p)n for a small, local probability, p , of breakup per unit of time. Estimated values of p remain nearly constant over many orders of magnitude of sample window size and over a range of simulation durations, n . This constant probability predicts a gradually falling values of the surviving proportions of such systems, f(n), which approach zero as n approaches infinity.