On the Figure-of-Eight and Nearby Solutions of the Three-Body Problem
(2025) In Bachelor’s Theses in Mathematical Sciences NUMK11 20251Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- The figure-of-eight solution is unusual in being a stable solution of the planar three-body problem. To study its stability, we first implement a numerical integrator, the Runge-Kutta-Fehlberg method, and demonstrate that it is accurate enough to integrate the figure-of-eight for thousands of periods. To examine what effect initial perturbations have on it, we define two perturbation schemes, one in the entire 12D phase space and one constrained to a 2D subset. We also define two departure metrics, one considering the topology of the orbit via the middle sequence and the other the shape of the orbit via the Fréchet distance metric, and find that these are equivalent for small perturbations. We find patterns in the departure times and... (More)
- The figure-of-eight solution is unusual in being a stable solution of the planar three-body problem. To study its stability, we first implement a numerical integrator, the Runge-Kutta-Fehlberg method, and demonstrate that it is accurate enough to integrate the figure-of-eight for thousands of periods. To examine what effect initial perturbations have on it, we define two perturbation schemes, one in the entire 12D phase space and one constrained to a 2D subset. We also define two departure metrics, one considering the topology of the orbit via the middle sequence and the other the shape of the orbit via the Fréchet distance metric, and find that these are equivalent for small perturbations. We find patterns in the departure times and explain these with references to dynamical features, including deflections, binary collisions, and escaping solutions. Furthermore, we use Monte Carlo simulation to estimate the internal radius of the stability region, but conclude that more integration time is necessary to produce statistically significant results. (Less)
- Popular Abstract
- The understanding of gravity we have today is easy to take for granted. It is used in ballistics calculation, predicting solar eclipses, sending scientific instruments to other planets, and much more. However, before Newton published his theory of gravity in 1687, it was anything but obvious that the force keeping our feet on the ground was the same force keeping the earth anchored to the sun. In the same publication, he also solved what we today call the two-body problem, by describing how to take the positions and velocities of two bodies (planets, stars, etc.) and predict their positions and velocities at any time in the future.
The key insight was that the problem was deterministic; knowing the state of the bodies at one time should... (More) - The understanding of gravity we have today is easy to take for granted. It is used in ballistics calculation, predicting solar eclipses, sending scientific instruments to other planets, and much more. However, before Newton published his theory of gravity in 1687, it was anything but obvious that the force keeping our feet on the ground was the same force keeping the earth anchored to the sun. In the same publication, he also solved what we today call the two-body problem, by describing how to take the positions and velocities of two bodies (planets, stars, etc.) and predict their positions and velocities at any time in the future.
The key insight was that the problem was deterministic; knowing the state of the bodies at one time should be enough to be able to predict their movements for all time, since there is no randomness. This idea motivated mathematicians to also try to solve the three-body problem, but this would prove to be much more difficult. It was revealed why when the mathematician Henri Poincaré discovered that the problem was chaotic, meaning that even very similar starting conditions can turn into very different behavior over time. This is the same property that makes it impossible to accurately forecast the weather far into the future.
Even though the three-body problem lacks a general solution, the movement of the bodies can still be simulated by computers. This way, many individual solutions have been found, and some of these repeat infinitely. One of these is the figure-of-eight solution, discovered in 2000, so named because the bodies follow each other on a figure-of-eight-shaped path. It is special in being one of the rare stable solutions. That means that the initial positions or velocities could be very slightly disturbed and still cause the bodies to follow similar paths -- thus making the figure-of-eight solution a stable island in a sea of chaos. In theory, it is therefore possible that a figure-of-eight configuration of three planets or three stars exists somewhere in the universe.
These ideas are developed further in this thesis, by perturbing the orbit and seeing if the figure-of-eight breaks down. One of the main questions of interest is how large the perturbations can be before they start to break the figure-of-eight, but we will also explore how quickly the figure-of-eight breaks depending on the perturbation. The implication is that even inexact astronomical observations could be used to determine if three planets or stars follow a figure-of-eight orbit, which is exciting. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9198240
- author
- Henriksson, Truls LU
- supervisor
- organization
- course
- NUMK11 20251
- year
- 2025
- type
- M2 - Bachelor Degree
- subject
- keywords
- three-body problem, ordinary differential equations, numerical integration, figure of eight, stability, chaos, monte carlo simulation, frechet distance
- publication/series
- Bachelor’s Theses in Mathematical Sciences
- report number
- LUNFNA-4063-2025
- ISSN
- 1654-6229
- other publication id
- 2025:K16
- language
- English
- id
- 9198240
- date added to LUP
- 2025-09-01 13:49:14
- date last changed
- 2025-09-01 13:49:14
@misc{9198240,
abstract = {{The figure-of-eight solution is unusual in being a stable solution of the planar three-body problem. To study its stability, we first implement a numerical integrator, the Runge-Kutta-Fehlberg method, and demonstrate that it is accurate enough to integrate the figure-of-eight for thousands of periods. To examine what effect initial perturbations have on it, we define two perturbation schemes, one in the entire 12D phase space and one constrained to a 2D subset. We also define two departure metrics, one considering the topology of the orbit via the middle sequence and the other the shape of the orbit via the Fréchet distance metric, and find that these are equivalent for small perturbations. We find patterns in the departure times and explain these with references to dynamical features, including deflections, binary collisions, and escaping solutions. Furthermore, we use Monte Carlo simulation to estimate the internal radius of the stability region, but conclude that more integration time is necessary to produce statistically significant results.}},
author = {{Henriksson, Truls}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor’s Theses in Mathematical Sciences}},
title = {{On the Figure-of-Eight and Nearby Solutions of the Three-Body Problem}},
year = {{2025}},
}