## Recommended Citation

Babbs, Charles F., "A fractal art approach to the three-body problem" (2024). *Weldon School of Biomedical Engineering Faculty Working Papers.* Paper 34.

https://docs.lib.purdue.edu/bmewp/34

## Date of this Version

2-7-2024

## Keywords

acceleration, collision, dipole, Earth, ejection, escape, gravity, image, Liebovitch, mass, Moon, Newton, orbit, planar, satellite, scale, self-similar, simulation, Sun, trajectory, Valtonen

## Abstract

This preliminary study explores a new search strategy for identifying relatively stable vs. unstable solutions to the planar three-body problem in astrophysics, starting from the perspective of computer-generated art. Here classical Newtonian accelerations, speeds, and positions of all three bodies in a fixed plane are calculated. All three bodies are stationary at time zero, and the fate of the system is classified as reflecting either a bound stable orbit, a likely collision, or the ejection of one body. The initial position of one of the three bodies is varied in the image plane, and the outcome coded as one of three colors, to produce a complex image of rings, defining either stable orbits, collisions, or ejection events. The nested, randomly interspersed, non-overlapping, both thick and thin rings resemble the rings of the planet Saturn seen up-close by a passing spacecraft. However, the rings are not concentric. Instead, they are similar to the field lines around an electric dipole. Since such field lines converge at the origin, detailed measurements of the ring density per unit length are possible either along a 45-degree line or along a horizontal line close to the origin. These measurements reveal a seemingly infinite number of rings of decreasing thicknesses over linear scales spanning 16 orders of magnitude. Such self-similar ring patterns at progressively smaller scales represent a new type of fractal, embedded in the classical three-body problem of astrophysics.